Average Error: 1.0 → 0.0
Time: 21.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt[3]{\left(\left(\frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}}\right) \cdot \left(\frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}}\right)\right) \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\sqrt[3]{\left(\left(\frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}}\right) \cdot \left(\frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{\pi - \left(v \cdot v\right) \cdot \pi}}\right)\right) \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r8932919 = 4.0;
        double r8932920 = 3.0;
        double r8932921 = atan2(1.0, 0.0);
        double r8932922 = r8932920 * r8932921;
        double r8932923 = 1.0;
        double r8932924 = v;
        double r8932925 = r8932924 * r8932924;
        double r8932926 = r8932923 - r8932925;
        double r8932927 = r8932922 * r8932926;
        double r8932928 = 2.0;
        double r8932929 = 6.0;
        double r8932930 = r8932929 * r8932925;
        double r8932931 = r8932928 - r8932930;
        double r8932932 = sqrt(r8932931);
        double r8932933 = r8932927 * r8932932;
        double r8932934 = r8932919 / r8932933;
        return r8932934;
}
↓
double f(double v) {
        double r8932935 = 1.3333333333333333;
        double r8932936 = sqrt(r8932935);
        double r8932937 = atan2(1.0, 0.0);
        double r8932938 = v;
        double r8932939 = r8932938 * r8932938;
        double r8932940 = r8932939 * r8932937;
        double r8932941 = r8932937 - r8932940;
        double r8932942 = sqrt(r8932941);
        double r8932943 = r8932936 / r8932942;
        double r8932944 = r8932943 * r8932943;
        double r8932945 = r8932944 * r8932944;
        double r8932946 = r8932935 / r8932941;
        double r8932947 = r8932945 * r8932946;
        double r8932948 = cbrt(r8932947);
        double r8932949 = -6.0;
        double r8932950 = 2.0;
        double r8932951 = fma(r8932949, r8932939, r8932950);
        double r8932952 = sqrt(r8932951);
        double r8932953 = r8932948 / r8932952;
        return r8932953;
}

Error

Bits error versus v

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. Simplified0.0

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

  4. Applied add-cbrt-cube1.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied times-frac0.0

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

  12. Applied add-sqr-sqrt0.0

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

  13. Applied add-sqr-sqrt0.0

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

  14. Applied times-frac0.0

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

  15. Applied swap-sqr0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))