Average Error: 1.0 → 0.0
Time: 25.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\sqrt[3]{\sqrt{\frac{4}{3}}} \cdot \sqrt[3]{\sqrt{\frac{4}{3}}}}{\pi}}{\sqrt{\sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2} \cdot \sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2}}} \cdot \frac{\frac{\sqrt[3]{\sqrt{\frac{4}{3}}}}{\frac{1 - v \cdot v}{\sqrt{\frac{4}{3}}}}}{\sqrt{\sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2}}}\]
\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{\sqrt[3]{\sqrt{\frac{4}{3}}} \cdot \sqrt[3]{\sqrt{\frac{4}{3}}}}{\pi}}{\sqrt{\sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2} \cdot \sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2}}} \cdot \frac{\frac{\sqrt[3]{\sqrt{\frac{4}{3}}}}{\frac{1 - v \cdot v}{\sqrt{\frac{4}{3}}}}}{\sqrt{\sqrt[3]{\left(v \cdot v\right) \cdot -6 + 2}}}
double f(double v) {
        double r14927074 = 4.0;
        double r14927075 = 3.0;
        double r14927076 = atan2(1.0, 0.0);
        double r14927077 = r14927075 * r14927076;
        double r14927078 = 1.0;
        double r14927079 = v;
        double r14927080 = r14927079 * r14927079;
        double r14927081 = r14927078 - r14927080;
        double r14927082 = r14927077 * r14927081;
        double r14927083 = 2.0;
        double r14927084 = 6.0;
        double r14927085 = r14927084 * r14927080;
        double r14927086 = r14927083 - r14927085;
        double r14927087 = sqrt(r14927086);
        double r14927088 = r14927082 * r14927087;
        double r14927089 = r14927074 / r14927088;
        return r14927089;
}
↓
double f(double v) {
        double r14927090 = 1.3333333333333333;
        double r14927091 = sqrt(r14927090);
        double r14927092 = cbrt(r14927091);
        double r14927093 = r14927092 * r14927092;
        double r14927094 = atan2(1.0, 0.0);
        double r14927095 = r14927093 / r14927094;
        double r14927096 = v;
        double r14927097 = r14927096 * r14927096;
        double r14927098 = -6.0;
        double r14927099 = r14927097 * r14927098;
        double r14927100 = 2.0;
        double r14927101 = r14927099 + r14927100;
        double r14927102 = cbrt(r14927101);
        double r14927103 = r14927102 * r14927102;
        double r14927104 = sqrt(r14927103);
        double r14927105 = r14927095 / r14927104;
        double r14927106 = 1.0;
        double r14927107 = r14927106 - r14927097;
        double r14927108 = r14927107 / r14927091;
        double r14927109 = r14927092 / r14927108;
        double r14927110 = sqrt(r14927102);
        double r14927111 = r14927109 / r14927110;
        double r14927112 = r14927105 * r14927111;
        return r14927112;
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/l*0.0

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

  6. Simplified0.0

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

  8. Applied add-cube-cbrt1.0

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

  9. Applied sqrt-prod1.0

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

  10. Applied *-un-lft-identity1.0

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

  11. Applied *-un-lft-identity1.0

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

  12. Applied distribute-rgt-out--1.0

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

  13. Applied times-frac1.0

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

  14. Applied add-cube-cbrt1.0

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

  15. Applied times-frac0.0

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

  16. Applied times-frac0.0

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

  17. Final simplification0.0

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

Reproduce

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