ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 87.5% → 97.0%
Time: 8.6s
Alternatives: 6
Speedup: 207.0×

Specification

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\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

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 6 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: 97.0% accurate, 0.3× speedup?

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-218}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(x \cdot \left({\varepsilon}^{3} \cdot 10\right) + 5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Derivation
  1. Split input into 3 regimes

  2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -2.0000000000000001e-218

    1. Initial program 99.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5} + \left(\left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
    3. Step-by-step derivation

      1. +-commutative100.0%

        \[\leadsto {\varepsilon}^{5} + \color{blue}{\left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x\right)} \]

      2. *-commutative100.0%

        \[\leadsto {\varepsilon}^{5} + \left(\color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x\right) \]

      3. unpow2100.0%

        \[\leadsto {\varepsilon}^{5} + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x\right) \]

      4. associate-*l*100.0%

        \[\leadsto {\varepsilon}^{5} + \left(\color{blue}{x \cdot \left(x \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)\right)} + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x\right) \]

      5. *-commutative100.0%

        \[\leadsto {\varepsilon}^{5} + \left(x \cdot \left(x \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)\right) + \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right)}\right) \]

      6. distribute-lft-out100.0%

        \[\leadsto {\varepsilon}^{5} + \color{blue}{x \cdot \left(x \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right)\right)} \]

    4. Simplified100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5} + x \cdot \left(x \cdot \left({\varepsilon}^{3} \cdot 10\right) + 5 \cdot {\varepsilon}^{4}\right)} \]

    if -2.0000000000000001e-218 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

    1. Initial program 86.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
    3. Step-by-step derivation

      1. distribute-lft1-in99.9%

        \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      2. metadata-eval99.9%

        \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. *-commutative99.9%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

    4. Simplified99.9%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]

    5. Taylor expanded in eps around 0 99.9%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
    6. Step-by-step derivation

      1. associate-*r*99.9%

        \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{4}} \]

      2. *-commutative99.9%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      3. associate-*r*99.9%

        \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

    7. Simplified99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

    if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

    1. Initial program 91.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification99.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-218}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(x \cdot \left({\varepsilon}^{3} \cdot 10\right) + 5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.3× speedup?

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-218} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -2.0000000000000001e-218 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

    1. Initial program 94.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    if -2.0000000000000001e-218 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

    1. Initial program 86.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
    3. Step-by-step derivation

      1. distribute-lft1-in99.9%

        \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      2. metadata-eval99.9%

        \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. *-commutative99.9%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

    4. Simplified99.9%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]

    5. Taylor expanded in eps around 0 99.9%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
    6. Step-by-step derivation

      1. associate-*r*99.9%

        \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{4}} \]

      2. *-commutative99.9%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      3. associate-*r*99.9%

        \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

    7. Simplified99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification99.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -2 \cdot 10^{-218} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 3: 95.2% accurate, 1.9× speedup?

\[\begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-59} \lor \neg \left(x \leq 8.8 \cdot 10^{-15}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if x < -7.59999999999999966e-59 or 8.79999999999999942e-15 < x

    1. Initial program 36.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 93.4%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
    3. Step-by-step derivation

      1. distribute-lft1-in93.4%

        \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      2. metadata-eval93.4%

        \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. associate-*l*93.4%

        \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

    4. Simplified93.4%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

    if -7.59999999999999966e-59 < x < 8.79999999999999942e-15

    1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification97.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-59} \lor \neg \left(x \leq 8.8 \cdot 10^{-15}\right):\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 4: 95.1% accurate, 1.9× speedup?

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-60} \lor \neg \left(x \leq 8.8 \cdot 10^{-15}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if x < -1.01999999999999994e-60 or 8.79999999999999942e-15 < x

    1. Initial program 36.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 93.4%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4}} \]
    3. Step-by-step derivation

      1. distribute-lft1-in93.4%

        \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      2. metadata-eval93.4%

        \[\leadsto \left(\color{blue}{5} \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. *-commutative93.4%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

    4. Simplified93.4%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4}} \]

    5. Taylor expanded in eps around 0 93.4%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
    6. Step-by-step derivation

      1. associate-*r*93.4%

        \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{4}} \]

      2. *-commutative93.4%

        \[\leadsto \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4} \]

      3. associate-*r*93.6%

        \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

    7. Simplified93.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

    if -1.01999999999999994e-60 < x < 8.79999999999999942e-15

    1. Initial program 99.3%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification97.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-60} \lor \neg \left(x \leq 8.8 \cdot 10^{-15}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 86.5% accurate, 2.0× speedup?

\[{\varepsilon}^{5} \]
Derivation

  1. Initial program 87.5%

    \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

  2. Taylor expanded in x around 0 86.1%

    \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

  3. Final simplification86.1%

    \[\leadsto {\varepsilon}^{5} \]

Alternative 6: 70.5% accurate, 207.0× speedup?

\[0 \]
Derivation

  1. Initial program 87.5%

    \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
  2. Step-by-step derivation

    1. add-sqr-sqrt73.1%

      \[\leadsto \color{blue}{\sqrt{{\left(x + \varepsilon\right)}^{5}} \cdot \sqrt{{\left(x + \varepsilon\right)}^{5}}} - {x}^{5} \]

    2. sqrt-unprod69.2%

      \[\leadsto \color{blue}{\sqrt{{\left(x + \varepsilon\right)}^{5} \cdot {\left(x + \varepsilon\right)}^{5}}} - {x}^{5} \]

    3. pow-prod-up68.1%

      \[\leadsto \sqrt{\color{blue}{{\left(x + \varepsilon\right)}^{\left(5 + 5\right)}}} - {x}^{5} \]

    4. metadata-eval68.1%

      \[\leadsto \sqrt{{\left(x + \varepsilon\right)}^{\color{blue}{10}}} - {x}^{5} \]

  3. Applied egg-rr68.1%

    \[\leadsto \color{blue}{\sqrt{{\left(x + \varepsilon\right)}^{10}}} - {x}^{5} \]
  4. Step-by-step derivation

    1. sqrt-pow187.5%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{\left(\frac{10}{2}\right)}} - {x}^{5} \]

    2. metadata-eval87.5%

      \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{5}} - {x}^{5} \]

    3. sqr-pow43.4%

      \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)} \cdot {\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}} - {x}^{5} \]

    4. pow243.4%

      \[\leadsto \color{blue}{{\left({\left(x + \varepsilon\right)}^{\left(\frac{5}{2}\right)}\right)}^{2}} - {x}^{5} \]

    5. metadata-eval43.4%

      \[\leadsto {\left({\left(x + \varepsilon\right)}^{\color{blue}{2.5}}\right)}^{2} - {x}^{5} \]

  5. Applied egg-rr43.4%

    \[\leadsto \color{blue}{{\left({\left(x + \varepsilon\right)}^{2.5}\right)}^{2}} - {x}^{5} \]

  6. Taylor expanded in x around inf 73.5%

    \[\leadsto \color{blue}{{x}^{5} + -1 \cdot {x}^{5}} \]
  7. Step-by-step derivation

    1. distribute-rgt1-in73.5%

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot {x}^{5}} \]

    2. metadata-eval73.5%

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

    3. mul0-lft73.5%

      \[\leadsto \color{blue}{0} \]

  8. Simplified73.5%

    \[\leadsto \color{blue}{0} \]

  9. Final simplification73.5%

    \[\leadsto 0 \]

Reproduce

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herbie shell --seed 2023167 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))