Average Error: 1.0 → 0.0
Time: 6.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot e^{\log \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}\]
\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)}}
↓
\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot e^{\log \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}
double code(double v) {
	return (4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v))))));
}
↓
double code(double v) {
	return (((4.0 / ((3.0 * ((double) M_PI)) * (pow(1.0, 3.0) - pow((v * v), 3.0)))) / sqrt((2.0 - (6.0 * (v * v))))) * exp(log(((1.0 * 1.0) + (((v * v) * (v * v)) + (1.0 * (v * v)))))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

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

  3. Applied add-exp-log1.0

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

  4. Applied add-exp-log1.0

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

  5. Applied add-exp-log1.0

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

  6. Applied add-exp-log1.0

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

  7. Applied prod-exp1.0

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

  8. Applied prod-exp1.0

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

  9. Applied prod-exp0.0

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

  10. Simplified0.0

    \[\leadsto \frac{4}{e^{\color{blue}{\log \left(\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}}}\]
  11. Using strategy rm

  12. Applied flip3--0.0

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

  13. Applied associate-*r/0.0

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

  14. Applied associate-*l/0.0

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

  15. Applied log-div0.0

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

  16. Applied exp-diff0.0

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

  17. Applied associate-/r/0.0

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

  18. Simplified0.0

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

  19. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))