Average Error: 1.0 → 0.0
Time: 3.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - \sqrt[3]{{\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}}\]
\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 - v \cdot v\right)}}{\sqrt{2 - \sqrt[3]{{\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}}
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)) * (1.0 - (v * v)))) / sqrt((2.0 - cbrt(pow((6.0 * (v * v)), 3.0)))));
}

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 associate-/r*0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied cbrt-unprod0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))