Average Error: 0.0 → 0.0
Time: 1.3m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\sqrt{2}}{4} \cdot \left(\left(\left(\left(\left(1 - \left(v \cdot 3\right) \cdot v\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \frac{\sqrt{2}}{4}\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{\frac{\sqrt{2}}{4} \cdot \left(\left(\left(\left(\left(1 - \left(v \cdot 3\right) \cdot v\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \frac{\sqrt{2}}{4}\right)}
double f(double v) {
        double r7118040 = 2.0;
        double r7118041 = sqrt(r7118040);
        double r7118042 = 4.0;
        double r7118043 = r7118041 / r7118042;
        double r7118044 = 1.0;
        double r7118045 = 3.0;
        double r7118046 = v;
        double r7118047 = r7118046 * r7118046;
        double r7118048 = r7118045 * r7118047;
        double r7118049 = r7118044 - r7118048;
        double r7118050 = sqrt(r7118049);
        double r7118051 = r7118043 * r7118050;
        double r7118052 = r7118044 - r7118047;
        double r7118053 = r7118051 * r7118052;
        return r7118053;
}
↓
double f(double v) {
        double r7118054 = 2.0;
        double r7118055 = sqrt(r7118054);
        double r7118056 = 4.0;
        double r7118057 = r7118055 / r7118056;
        double r7118058 = 1.0;
        double r7118059 = v;
        double r7118060 = 3.0;
        double r7118061 = r7118059 * r7118060;
        double r7118062 = r7118061 * r7118059;
        double r7118063 = r7118058 - r7118062;
        double r7118064 = r7118059 * r7118059;
        double r7118065 = r7118058 - r7118064;
        double r7118066 = r7118065 * r7118065;
        double r7118067 = r7118063 * r7118066;
        double r7118068 = sqrt(r7118063);
        double r7118069 = r7118068 * r7118065;
        double r7118070 = r7118067 * r7118069;
        double r7118071 = r7118070 * r7118057;
        double r7118072 = r7118071 * r7118057;
        double r7118073 = r7118057 * r7118072;
        double r7118074 = cbrt(r7118073);
        return r7118074;
}

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 associate-*l*0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied add-cbrt-cube1.0

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

  10. Applied cbrt-undiv0.0

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

  11. Applied cbrt-unprod0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))