subtraction fraction

Percentage Accurate: 100.0% → 100.0%
Time: 3.9s
Alternatives: 5
Speedup: TODO×

Specification

?
\[\frac{-\left(f + n\right)}{f - n} \]

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 5 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1?

\[\frac{1}{\frac{n - f}{n + f}} \]
Derivation

  1. Initial program 100.0%

    \[\frac{-\left(f + n\right)}{f - n} \]
  2. Step-by-step derivation

    1. neg-mul-1100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

    2. *-commutative100.0%

      \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

    3. associate-/l*100.0%

      \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

    4. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

    5. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

    6. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

    7. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

    8. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

    9. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

    10. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

    11. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

    12. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

    13. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

    14. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

    15. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

    16. unsub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

    17. remove-double-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

    18. +-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

    19. sub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

    20. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

    21. /-rgt-identity100.0%

      \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
  4. Step-by-step derivation

    1. add-cbrt-cube100.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{f + n}{n - f} \cdot \frac{f + n}{n - f}\right) \cdot \frac{f + n}{n - f}}} \]

    2. pow3100.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]

  5. Applied egg-rr100.0%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]
  6. Step-by-step derivation

    1. rem-cbrt-cube100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

    2. clear-num100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

  7. Applied egg-rr100.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]

  8. Final simplification100.0%

    \[\leadsto \frac{1}{\frac{n - f}{n + f}} \]

Alternative 2?

\[\begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{+79}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if n < -1.22e19 or 1.5499999999999999e79 < n

    1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
    2. Step-by-step derivation

      1. neg-mul-1100.0%

        \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

      2. *-commutative100.0%

        \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

      3. associate-/l*100.0%

        \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

      4. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

      5. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

      6. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

      7. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

      8. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

      9. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

      10. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

      11. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

      12. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

      13. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

      14. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

      15. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

      16. unsub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

      17. remove-double-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

      18. +-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

      19. sub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

      20. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

      21. /-rgt-identity100.0%

        \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

    3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

    4. Taylor expanded in f around 0 89.4%

      \[\leadsto \color{blue}{1} \]

    if -1.22e19 < n < 1.5499999999999999e79

    1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
    2. Step-by-step derivation

      1. neg-mul-1100.0%

        \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

      2. *-commutative100.0%

        \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

      3. associate-/l*100.0%

        \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

      4. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

      5. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

      6. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

      7. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

      8. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

      9. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

      10. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

      11. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

      12. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

      13. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

      14. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

      15. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

      16. unsub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

      17. remove-double-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

      18. +-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

      19. sub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

      20. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

      21. /-rgt-identity100.0%

        \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

    3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

    4. Taylor expanded in n around 0 76.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{n}{f} - 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification82.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.55 \cdot 10^{+79}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3?

\[\frac{n + f}{n - f} \]
Derivation

  1. Initial program 100.0%

    \[\frac{-\left(f + n\right)}{f - n} \]
  2. Step-by-step derivation

    1. neg-mul-1100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

    2. *-commutative100.0%

      \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

    3. associate-/l*100.0%

      \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

    4. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

    5. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

    6. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

    7. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

    8. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

    9. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

    10. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

    11. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

    12. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

    13. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

    14. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

    15. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

    16. unsub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

    17. remove-double-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

    18. +-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

    19. sub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

    20. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

    21. /-rgt-identity100.0%

      \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

  4. Final simplification100.0%

    \[\leadsto \frac{n + f}{n - f} \]

Alternative 4?

\[\begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+77}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if n < -3.4e18 or 3.8999999999999998e77 < n

    1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
    2. Step-by-step derivation

      1. neg-mul-1100.0%

        \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

      2. *-commutative100.0%

        \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

      3. associate-/l*100.0%

        \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

      4. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

      5. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

      6. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

      7. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

      8. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

      9. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

      10. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

      11. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

      12. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

      13. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

      14. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

      15. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

      16. unsub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

      17. remove-double-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

      18. +-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

      19. sub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

      20. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

      21. /-rgt-identity100.0%

        \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

    3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

    4. Taylor expanded in f around 0 89.4%

      \[\leadsto \color{blue}{1} \]

    if -3.4e18 < n < 3.8999999999999998e77

    1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
    2. Step-by-step derivation

      1. neg-mul-1100.0%

        \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

      2. *-commutative100.0%

        \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

      3. associate-/l*100.0%

        \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

      4. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

      5. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

      6. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

      7. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

      8. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

      9. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

      10. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

      11. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

      12. associate-/l*100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

      13. *-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

      14. neg-mul-1100.0%

        \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

      15. div-sub100.0%

        \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

      16. unsub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

      17. remove-double-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

      18. +-commutative100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

      19. sub-neg100.0%

        \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

      20. metadata-eval100.0%

        \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

      21. /-rgt-identity100.0%

        \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

    3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

    4. Taylor expanded in f around inf 75.3%

      \[\leadsto \color{blue}{-1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification81.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.4 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+77}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5?

\[-1 \]
Derivation

  1. Initial program 100.0%

    \[\frac{-\left(f + n\right)}{f - n} \]
  2. Step-by-step derivation

    1. neg-mul-1100.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]

    2. *-commutative100.0%

      \[\leadsto \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]

    3. associate-/l*100.0%

      \[\leadsto \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]

    4. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]

    5. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]

    6. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]

    7. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]

    8. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]

    9. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]

    10. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]

    11. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]

    12. associate-/l*100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]

    13. *-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]

    14. neg-mul-1100.0%

      \[\leadsto \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]

    15. div-sub100.0%

      \[\leadsto \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]

    16. unsub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]

    17. remove-double-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]

    18. +-commutative100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]

    19. sub-neg100.0%

      \[\leadsto \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]

    20. metadata-eval100.0%

      \[\leadsto \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]

    21. /-rgt-identity100.0%

      \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

  4. Taylor expanded in f around inf 47.5%

    \[\leadsto \color{blue}{-1} \]

  5. Final simplification47.5%

    \[\leadsto -1 \]

Reproduce

?
herbie shell --seed 2023166 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))