Average Error: 1.0 → 0.0
Time: 2.6s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{{\left(\frac{4}{3 \cdot \left(\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}\right)\right)}\right)}^{3}}\]
\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)}}
↓
\sqrt[3]{{\left(\frac{4}{3 \cdot \left(\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}\right)\right)}\right)}^{3}}
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 ((double) cbrt(((double) pow((4.0 / ((double) (3.0 * ((double) (((double) M_PI) * ((double) (((double) (1.0 - ((double) (v * v)))) * ((double) sqrt(((double) (2.0 - ((double) (v * ((double) (v * 6.0))))))))))))))), 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 Error: 1.0 bits

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

  2. SimplifiedError: 1.0 bits

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

  4. Applied add-cbrt-cubeError: 1.0 bits

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

  5. Applied add-cbrt-cubeError: 1.0 bits

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

  6. Applied cbrt-unprodError: 1.0 bits

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

  7. Applied add-cbrt-cubeError: 1.6 bits

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

  8. Applied cbrt-unprodError: 1.0 bits

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

  9. Applied add-cbrt-cubeError: 1.6 bits

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

  10. Applied cbrt-unprodError: 1.0 bits

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

  11. Applied add-cbrt-cubeError: 1.0 bits

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

  12. Applied cbrt-undivError: 0.0 bits

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

  13. SimplifiedError: 0.0 bits

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

  14. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020200 
(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)))))))