Average Error: 1.0 → 0.0
Time: 29.5min
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\pi}}{\left(\left(1 - v \cdot v\right) \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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}{\pi}}{\left(\left(1 - v \cdot v\right) \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double code(double v) {
	return (4.0 / ((double) (((double) (((double) (3.0 * ((double) M_PI))) * ((double) (1.0 - ((double) (v * v)))))) * ((double) sqrt(((double) (2.0 - ((double) (6.0 * ((double) (v * v)))))))))));
}
↓
double code(double v) {
	return ((4.0 / ((double) M_PI)) / ((double) (((double) (((double) (1.0 - ((double) (v * v)))) * 3.0)) * ((double) sqrt(((double) (2.0 - ((double) (6.0 * ((double) (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-cbrt-cube1.0

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

  4. Applied add-cbrt-cube1.0

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

  5. Applied cbrt-undiv1.0

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

  6. Simplified0.0

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

  8. Applied *-un-lft-identity0.0

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

  9. Applied times-frac0.0

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

  10. Applied unpow-prod-down0.0

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

  11. Applied cbrt-prod1.0

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

  12. Simplified1.0

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

  13. Simplified0.0

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

  15. Applied frac-times0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2020182 
(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)))))))