Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%
Time: 4.6s
Alternatives: 6
Speedup: TODO×

Specification

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\[\begin{array}{l} t_0 := v \cdot v\\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - t_0\right)\right) \cdot \sqrt{2 - 6 \cdot t_0}} \end{array} \]

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?

\[\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Step-by-step derivation

    1. associate-/r*100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

    2. associate-*l*100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    4. metadata-eval100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    5. sub-neg100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

    6. *-commutative100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right)}} \]

    7. distribute-rgt-neg-in100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]

    8. metadata-eval100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
  4. Step-by-step derivation

    1. expm1-log1p-u98.5%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}\right)\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

    2. expm1-udef100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}\right)} - 1}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  5. Applied egg-rr100.0%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}\right)} - 1}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
  6. Step-by-step derivation

    1. expm1-def98.5%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}\right)\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

    2. expm1-log1p100.0%

      \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  7. Simplified100.0%

    \[\leadsto \frac{\color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  8. Final simplification100.0%

    \[\leadsto \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

Alternative 2?

\[\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Step-by-step derivation

    1. associate-/r*100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

    2. associate-*l*100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    4. metadata-eval100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    5. sub-neg100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

    6. *-commutative100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right)}} \]

    7. distribute-rgt-neg-in100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]

    8. metadata-eval100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

  4. Final simplification100.0%

    \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

Alternative 3?

\[\frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Step-by-step derivation

    1. associate-/r*100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

    2. associate-*l*100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    4. metadata-eval100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    5. sub-neg100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

    6. *-commutative100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right)}} \]

    7. distribute-rgt-neg-in100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]

    8. metadata-eval100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

  4. Taylor expanded in v around 0 98.2%

    \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
  5. Step-by-step derivation

    1. div-inv98.2%

      \[\leadsto \frac{\color{blue}{1.3333333333333333 \cdot \frac{1}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  6. Applied egg-rr98.2%

    \[\leadsto \frac{\color{blue}{1.3333333333333333 \cdot \frac{1}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  7. Final simplification98.2%

    \[\leadsto \frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

Alternative 4?

\[\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Step-by-step derivation

    1. associate-/r*100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

    2. associate-*l*100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    4. metadata-eval100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    5. sub-neg100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

    6. *-commutative100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right)}} \]

    7. distribute-rgt-neg-in100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]

    8. metadata-eval100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

  4. Taylor expanded in v around 0 98.2%

    \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  5. Final simplification98.2%

    \[\leadsto \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

Alternative 5?

\[1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

  2. Taylor expanded in v around 0 96.7%

    \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi}} \]

  3. Final simplification96.7%

    \[\leadsto 1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi} \]

Alternative 6?

\[\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2}} \]
Derivation

  1. Initial program 98.5%

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Step-by-step derivation

    1. associate-/r*100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

    2. associate-*l*100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    3. associate-/r*100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    4. metadata-eval100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

    5. sub-neg100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

    6. *-commutative100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(v \cdot v\right) \cdot 6}\right)}} \]

    7. distribute-rgt-neg-in100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]

    8. metadata-eval100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

  3. Simplified100.0%

    \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

  4. Taylor expanded in v around 0 98.2%

    \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

  5. Taylor expanded in v around 0 98.2%

    \[\leadsto \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{\color{blue}{2}}} \]

  6. Final simplification98.2%

    \[\leadsto \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2}} \]

Reproduce

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herbie shell --seed 2023166 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))