Average Error: 0.0 → 0.0
Time: 12.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r182898 = 2.0;
        double r182899 = sqrt(r182898);
        double r182900 = 4.0;
        double r182901 = r182899 / r182900;
        double r182902 = 1.0;
        double r182903 = 3.0;
        double r182904 = v;
        double r182905 = r182904 * r182904;
        double r182906 = r182903 * r182905;
        double r182907 = r182902 - r182906;
        double r182908 = sqrt(r182907);
        double r182909 = r182901 * r182908;
        double r182910 = r182902 - r182905;
        double r182911 = r182909 * r182910;
        return r182911;
}
↓
double f(double v) {
        double r182912 = 2.0;
        double r182913 = sqrt(r182912);
        double r182914 = 4.0;
        double r182915 = r182913 / r182914;
        double r182916 = 1.0;
        double r182917 = 3.0;
        double r182918 = v;
        double r182919 = r182918 * r182918;
        double r182920 = r182917 * r182919;
        double r182921 = r182916 - r182920;
        double r182922 = cbrt(r182921);
        double r182923 = fabs(r182922);
        double r182924 = sqrt(r182922);
        double r182925 = r182923 * r182924;
        double r182926 = r182915 * r182925;
        double r182927 = r182916 - r182919;
        double r182928 = r182926 * r182927;
        return r182928;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))