Average Error: 0.0 → 0.0
Time: 7.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot \left(\sqrt{2} \cdot \frac{1}{32}\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot \left(\sqrt{2} \cdot \frac{1}{32}\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}
double f(double v) {
        double r3067903 = 2.0;
        double r3067904 = sqrt(r3067903);
        double r3067905 = 4.0;
        double r3067906 = r3067904 / r3067905;
        double r3067907 = 1.0;
        double r3067908 = 3.0;
        double r3067909 = v;
        double r3067910 = r3067909 * r3067909;
        double r3067911 = r3067908 * r3067910;
        double r3067912 = r3067907 - r3067911;
        double r3067913 = sqrt(r3067912);
        double r3067914 = r3067906 * r3067913;
        double r3067915 = r3067907 - r3067910;
        double r3067916 = r3067914 * r3067915;
        return r3067916;
}
↓
double f(double v) {
        double r3067917 = 1.0;
        double r3067918 = v;
        double r3067919 = r3067918 * r3067918;
        double r3067920 = r3067917 - r3067919;
        double r3067921 = 3.0;
        double r3067922 = r3067919 * r3067921;
        double r3067923 = r3067917 - r3067922;
        double r3067924 = 2.0;
        double r3067925 = sqrt(r3067924);
        double r3067926 = 0.03125;
        double r3067927 = r3067925 * r3067926;
        double r3067928 = r3067923 * r3067927;
        double r3067929 = sqrt(r3067923);
        double r3067930 = r3067928 * r3067929;
        double r3067931 = cbrt(r3067930);
        double r3067932 = r3067920 * r3067931;
        return r3067932;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-undiv0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \left(\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot \left(\frac{1}{32} \cdot \sqrt{2}\right)\right)}} \cdot \left(1 - v \cdot v\right)\]

  9. Final simplification0.0

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot \left(\sqrt{2} \cdot \frac{1}{32}\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))