ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%
Time: 4.2s
Alternatives: 5
Speedup: 7.0×

Specification

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\[\frac{10}{1 - x \cdot x} \]

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: 99.6% accurate, 0.1× speedup?

\[\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \]
Derivation

  1. Initial program 87.8%

    \[\frac{10}{1 - x \cdot x} \]
  2. Step-by-step derivation

    1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

    2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

    3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

    4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

    5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

    6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

    7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

    8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

    9. fma-neg99.7%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

    10. metadata-eval99.7%

      \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

  4. Final simplification99.7%

    \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \]

Alternative 2: 13.5% accurate, 0.8× speedup?

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if (*.f64 x x) < 1

    1. Initial program 88.3%

      \[\frac{10}{1 - x \cdot x} \]
    2. Step-by-step derivation

      1. sub-neg88.3%

        \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative88.3%

        \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub088.3%

        \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-88.3%

        \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg88.3%

        \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-188.3%

        \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*88.3%

        \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval88.3%

        \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.6%

        \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.6%

        \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

    3. Simplified99.6%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

    4. Taylor expanded in x around 0 13.5%

      \[\leadsto \color{blue}{10} \]

    if 1 < (*.f64 x x)

    1. Initial program 87.1%

      \[\frac{10}{1 - x \cdot x} \]
    2. Step-by-step derivation

      1. sub-neg87.1%

        \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative87.1%

        \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub087.1%

        \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-87.1%

        \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg87.1%

        \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-187.1%

        \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*87.1%

        \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval87.1%

        \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.7%

        \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.7%

        \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

    3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

    4. Taylor expanded in x around inf 13.5%

      \[\leadsto \color{blue}{\frac{-10}{{x}^{2}}} \]
    5. Step-by-step derivation

      1. unpow213.5%

        \[\leadsto \frac{-10}{\color{blue}{x \cdot x}} \]

    6. Simplified13.5%

      \[\leadsto \color{blue}{\frac{-10}{x \cdot x}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \end{array} \]

Alternative 3: 87.8% accurate, 0.8× speedup?

\[\frac{1}{1 - x \cdot x} \cdot 10 \]
Derivation

  1. Initial program 87.8%

    \[\frac{10}{1 - x \cdot x} \]
  2. Step-by-step derivation

    1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

    2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

    3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

    4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

    5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

    6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

    7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

    8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

    9. fma-neg99.7%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

    10. metadata-eval99.7%

      \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
  4. Step-by-step derivation

    1. frac-2neg99.7%

      \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

    2. metadata-eval99.7%

      \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

    3. fma-udef87.8%

      \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

    4. distribute-neg-in87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

    5. metadata-eval87.8%

      \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

    6. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

    7. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

    8. div-inv87.8%

      \[\leadsto \color{blue}{10 \cdot \frac{1}{1 - x \cdot x}} \]

    9. *-commutative87.8%

      \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot 10} \]

  5. Applied egg-rr87.8%

    \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot 10} \]

  6. Final simplification87.8%

    \[\leadsto \frac{1}{1 - x \cdot x} \cdot 10 \]

Alternative 4: 87.8% accurate, 1.0× speedup?

\[\frac{10}{1 - x \cdot x} \]
Derivation

  1. Initial program 87.8%

    \[\frac{10}{1 - x \cdot x} \]

  2. Final simplification87.8%

    \[\leadsto \frac{10}{1 - x \cdot x} \]

Alternative 5: 9.5% accurate, 7.0× speedup?

\[10 \]
Derivation

  1. Initial program 87.8%

    \[\frac{10}{1 - x \cdot x} \]
  2. Step-by-step derivation

    1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

    2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

    3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

    4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

    5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

    6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

    7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

    8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

    9. fma-neg99.7%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

    10. metadata-eval99.7%

      \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

  4. Taylor expanded in x around 0 9.0%

    \[\leadsto \color{blue}{10} \]

  5. Final simplification9.0%

    \[\leadsto 10 \]

Reproduce

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herbie shell --seed 2023167 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, B"
  :precision binary64
  :pre (and (<= 0.999 x) (<= x 1.001))
  (/ 10.0 (- 1.0 (* x x))))