Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%
Time: 7.9s
Alternatives: 5
Speedup: 2.1×

Specification

?
\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
\begin{array}{l}

\\
\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
\end{array}

Sampling outcomes in binary64 precision:

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.

Initial Program: 98.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
\begin{array}{l}

\\
\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
\end{array}

Alternative 1: 100.0% accurate, 0.9× speedup?

\[\begin{array}{l} \\ \frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \end{array} \]
(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* 3.0 (PI))) (* (- 1.0 (* v v)) (sqrt (fma (* v v) -6.0 2.0)))))
\begin{array}{l}

\\
\frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}
\end{array}
Derivation

  1. Initial program 98.4%

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

    1. lift--.f64N/A

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

    2. sub-negN/A

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

    3. +-commutativeN/A

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

    4. lift-*.f64N/A

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

    5. *-commutativeN/A

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

    6. distribute-rgt-neg-inN/A

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

    7. lower-fma.f64N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

    8. metadata-eval98.4

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

  4. Applied rewrites98.4%

    \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
  5. Step-by-step derivation

    1. lift-/.f64N/A

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

    2. lift-*.f64N/A

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

    3. lift-*.f64N/A

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

    4. associate-*l*N/A

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

    5. associate-/r*N/A

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

    6. lower-/.f64N/A

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

    7. lower-/.f64N/A

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

    8. lift-fma.f64N/A

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

    9. *-commutativeN/A

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

    10. lift-fma.f64N/A

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

    11. lower-*.f64100.0

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

    12. lift-fma.f64N/A

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

    13. *-commutativeN/A

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

    14. lift-fma.f64100.0

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

  6. Applied rewrites100.0%

    \[\leadsto \color{blue}{\frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
  7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \frac{-1.3333333333333333}{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right) \cdot \mathsf{PI}\left(\right)} \end{array} \]
(FPCore (v)
 :precision binary64
 (/
  -1.3333333333333333
  (* (* (fma v v -1.0) (sqrt (fma (* v v) -6.0 2.0))) (PI))))
\begin{array}{l}

\\
\frac{-1.3333333333333333}{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right) \cdot \mathsf{PI}\left(\right)}
\end{array}
Derivation

  1. Initial program 98.4%

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

    1. lift--.f64N/A

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

    2. sub-negN/A

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

    3. +-commutativeN/A

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

    4. lift-*.f64N/A

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

    5. *-commutativeN/A

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

    6. distribute-rgt-neg-inN/A

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

    7. lower-fma.f64N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \mathsf{neg}\left(6\right), 2\right)}}} \]

    8. metadata-eval98.4

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

  4. Applied rewrites98.4%

    \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
  5. Step-by-step derivation

    1. lift-/.f64N/A

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

    2. lift-*.f64N/A

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

    3. associate-/r*N/A

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

    4. lift-fma.f64N/A

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

    5. *-commutativeN/A

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

    6. lift-fma.f64N/A

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

    7. div-invN/A

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

    8. lift-*.f64N/A

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

    9. lift-*.f64N/A

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

    10. associate-*l*N/A

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

    11. associate-/r*N/A

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

    12. metadata-evalN/A

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

    13. frac-timesN/A

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

  6. Applied rewrites100.0%

    \[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
  7. Step-by-step derivation

    1. lift-*.f64N/A

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

    2. *-commutativeN/A

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

    3. lift-*.f64N/A

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

    4. associate-*r*N/A

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

    5. *-commutativeN/A

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

    6. lift-*.f64N/A

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

    7. lower-*.f64100.0

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

    8. lift-*.f64N/A

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

    9. *-commutativeN/A

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

    10. lower-*.f64100.0

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

    11. lift-fma.f64N/A

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

    12. *-commutativeN/A

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

    13. lift-fma.f64100.0

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

  8. Applied rewrites100.0%

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

  10. Applied rewrites100.0%

    \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}\right) \cdot \mathsf{PI}\left(\right)}} \]
  11. Add Preprocessing

Alternative 3: 99.1% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \end{array} \]
(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* (PI) (sqrt (fma -6.0 (* v v) 2.0)))))
\begin{array}{l}

\\
\frac{1.3333333333333333}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
\end{array}
Derivation

  1. Initial program 98.4%

    \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Add Preprocessing

  4. Applied rewrites99.5%

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

  5. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
  6. Step-by-step derivation

    1. lower-PI.f6499.5

      \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

  7. Applied rewrites99.5%

    \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
  8. Add Preprocessing

Alternative 4: 99.0% accurate, 2.1× speedup?

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \end{array} \]
(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* (sqrt 2.0) (PI))))
\begin{array}{l}

\\
\frac{1.3333333333333333}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}
\end{array}
Derivation

  1. Initial program 98.4%

    \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Add Preprocessing

  4. Applied rewrites99.5%

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

  5. Taylor expanded in v around 0

    \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
  6. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]

    2. lower-*.f64N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]

    3. lower-sqrt.f64N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)} \]

    4. lower-PI.f6499.5

      \[\leadsto \frac{1.3333333333333333}{\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

  7. Applied rewrites99.5%

    \[\leadsto \frac{1.3333333333333333}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]
  8. Add Preprocessing

Alternative 5: 97.5% accurate, 2.1× speedup?

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333 \end{array} \]
(FPCore (v) :precision binary64 (* (/ (sqrt 0.5) (PI)) 1.3333333333333333))
\begin{array}{l}

\\
\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333
\end{array}
Derivation

  1. Initial program 98.4%

    \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
  2. Add Preprocessing

  3. Taylor expanded in v around 0

    \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
  4. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3}} \]

    2. lower-*.f64N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3}} \]

    3. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \cdot \frac{4}{3} \]

    4. lower-sqrt.f64N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3} \]

    5. lower-PI.f6498.0

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\mathsf{PI}\left(\right)}} \cdot 1.3333333333333333 \]

  5. Applied rewrites98.0%

    \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333} \]
  6. Add Preprocessing

Reproduce

?
herbie shell --seed 2024307 
(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)))))))