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Affine Residual Scaling, Forward Error and Tolerance Conventions
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batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
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1. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-3, intercept=1, approximation=-4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
2. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-2, intercept=2, approximation=-3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
3. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-1, intercept=3, approximation=-2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
4. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=4, approximation=-1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
5. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=1, intercept=5, approximation=0 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
6. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=6, approximation=1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
7. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=3, intercept=7, approximation=2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
8. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-3, intercept=8, approximation=3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
9. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
10. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-1, intercept=10, approximation=5 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
11. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=11, approximation=6 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
12. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=1, intercept=12, approximation=7 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
13. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=13, approximation=8 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
14. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=3, intercept=14, approximation=9 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
15. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-3, intercept=15, approximation=10 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
16. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-2, intercept=16, approximation=11 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
17. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-1, intercept=17, approximation=12 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
18. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=18, approximation=13 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
19. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=1, intercept=19, approximation=14 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
20. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=20, approximation=15 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
21. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=3, intercept=21, approximation=16 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
22. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-3, intercept=22, approximation=17 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
23. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-2, intercept=23, approximation=18 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
24. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=-1, intercept=24, approximation=19 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
25. An Affine Residual Equals Absolute Slope Times Forward Error
Use nonzero slope=2, intercept=25, approximation=20 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).
View answer and explanation
(2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities)
Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-residual-forward · candidate
26. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-3 and nonnegative absolute input error bound=1/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-3, 1/3, 1, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 1/3, 1, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
27. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-2 and nonnegative absolute input error bound=2/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-2, 2/3, 4/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 2/3, 4/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
28. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-1 and nonnegative absolute input error bound=3/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-1, 1, 1, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 1, 1, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
29. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=4/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 4/3, 8/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 4/3, 8/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
30. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=1 and nonnegative absolute input error bound=5/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(1, 5/3, 5/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 5/3, 5/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
31. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=6/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 2, 4, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 2, 4, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
32. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=3 and nonnegative absolute input error bound=7/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(3, 7/3, 7, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7/3, 7, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
33. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-3 and nonnegative absolute input error bound=8/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-3, 8/3, 8, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 8/3, 8, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
34. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-2 and nonnegative absolute input error bound=9/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-2, 3, 6, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 3, 6, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
35. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-1 and nonnegative absolute input error bound=10/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-1, 10/3, 10/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 10/3, 10/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
36. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=11/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 11/3, 22/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 11/3, 22/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
37. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=1 and nonnegative absolute input error bound=12/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(1, 4, 4, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 4, 4, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
38. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=13/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 13/3, 26/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 13/3, 26/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
39. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=3 and nonnegative absolute input error bound=14/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(3, 14/3, 14, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 14/3, 14, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
40. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-3, 5, 15, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 5, 15, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
41. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-2 and nonnegative absolute input error bound=16/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-2, 16/3, 32/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 16/3, 32/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
42. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-1 and nonnegative absolute input error bound=17/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-1, 17/3, 17/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 17/3, 17/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
43. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=18/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 6, 12, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 6, 12, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
44. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=1 and nonnegative absolute input error bound=19/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(1, 19/3, 19/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 19/3, 19/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
45. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=20/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 20/3, 40/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 20/3, 40/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
46. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=3 and nonnegative absolute input error bound=21/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(3, 7, 21, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7, 21, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
47. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-3 and nonnegative absolute input error bound=22/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-3, 22/3, 22, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 22/3, 22, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
48. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-2 and nonnegative absolute input error bound=23/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-2, 23/3, 46/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 23/3, 46/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
49. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=-1 and nonnegative absolute input error bound=24/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(-1, 8, 8, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 8, 8, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
50. An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope
Use slope=2 and nonnegative absolute input error bound=25/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).
View answer and explanation
(2, 25/3, 50/3, affine_output_absolute_error_bound)
Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 25/3, 50/3, affine_output_absolute_error_bound).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-affine-propagation · candidate
51. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=1, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
52. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=2, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
53. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=3, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
54. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=4, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
55. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=5, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
56. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=6, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
57. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=7, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
58. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=8, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
59. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=9, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
60. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=10, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
61. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=11, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
62. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=12, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
63. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=13, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
64. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=14, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
65. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=15, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
66. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=16, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
67. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=17, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
68. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=18, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
69. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=19, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
70. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=20, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
71. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=21, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
72. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=22, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
73. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=23, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
74. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=24, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
75. The Same Residual Can Correspond to Different Forward Errors
Use positive absolute residual=25, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).
View answer and explanation
(25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)
Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-conditioning-witness · candidate
76. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":1}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 1, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 1, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
77. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[2,2],"tolerance":[2,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
78. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[3,2],"tolerance":[3,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 3/2, 3/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 3/2, 3/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
79. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":4,"reference":-4}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 4, -4, 8, 8, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 4, -4, 8, 8, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
80. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":5}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 5, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 5, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
81. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[6,2],"tolerance":[6,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 3, 3, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 3, 3, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
82. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[7,2],"tolerance":[7,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 7/2, 7/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 7/2, 7/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
83. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":8,"reference":-8}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 8, -8, 16, 16, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 8, -8, 16, 16, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
84. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":9}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 9, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 9, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
85. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[10,2],"tolerance":[10,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 5, 5, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 5, 5, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
86. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[11,2],"tolerance":[11,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 11/2, 11/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 11/2, 11/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
87. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":12,"reference":-12}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 12, -12, 24, 24, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 12, -12, 24, 24, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
88. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":13}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 13, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 13, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
89. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[14,2],"tolerance":[14,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 7, 7, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 7, 7, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
90. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[15,2],"tolerance":[15,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 15/2, 15/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 15/2, 15/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
91. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":16,"reference":-16}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 16, -16, 32, 32, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 16, -16, 32, 32, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
92. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":17}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 17, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 17, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
93. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[18,2],"tolerance":[18,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 9, 9, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 9, 9, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
94. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[19,2],"tolerance":[19,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 19/2, 19/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 19/2, 19/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
95. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":20,"reference":-20}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 20, -20, 40, 40, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 20, -20, 40, 40, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
96. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":21}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 21, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 21, reference=0, relative_error=undefined, No).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
97. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=inclusive_tolerance, complete model={"error":[22,2],"tolerance":[22,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(inclusive_tolerance, 11, 11, Yes, error<=tolerance_is_true_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 11, 11, Yes, error<=tolerance_is_true_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
98. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=strict_tolerance, complete model={"error":[23,2],"tolerance":[23,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(strict_tolerance, 23/2, 23/2, No, error<tolerance_is_false_at_equality)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 23/2, 23/2, No, error<tolerance_is_false_at_equality).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
99. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=signed_vs_absolute, complete model={"approximation":24,"reference":-24}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(signed_vs_absolute, 24, -24, 48, 48, No, signed_error_and_absolute_error_preserve_different_information)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 24, -24, 48, 48, No, signed_error_and_absolute_error_preserve_different_information).
Reviewed step-by-step derivation
Numerical approximation and error · qf-mod50-numerr-boundary · candidate
100. Zero Reference and Equal Tolerances Require Explicit Error Conventions
Use mode=zero_reference, complete model={"approximation":25}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).
The four modes have5/5/5/7 fields. Strict and inclusive modes have equal arity but opposite results at equality. Relative error is undefined at a zero reference.
View answer and explanation
(zero_reference, 25, reference=0, relative_error=undefined, No)
Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 25, reference=0, relative_error=undefined, No).
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Numerical approximation and error · qf-mod50-numerr-boundary · candidate
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