Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)
double f(double v) {
        double r321900 = 2.0;
        double r321901 = sqrt(r321900);
        double r321902 = 4.0;
        double r321903 = r321901 / r321902;
        double r321904 = 1.0;
        double r321905 = 3.0;
        double r321906 = v;
        double r321907 = r321906 * r321906;
        double r321908 = r321905 * r321907;
        double r321909 = r321904 - r321908;
        double r321910 = sqrt(r321909);
        double r321911 = r321903 * r321910;
        double r321912 = r321904 - r321907;
        double r321913 = r321911 * r321912;
        return r321913;
}
↓
double f(double v) {
        double r321914 = 2.0;
        double r321915 = sqrt(r321914);
        double r321916 = 4.0;
        double r321917 = r321915 / r321916;
        double r321918 = 1.0;
        double r321919 = 3.0;
        double r321920 = v;
        double r321921 = r321920 * r321920;
        double r321922 = r321919 * r321921;
        double r321923 = r321918 - r321922;
        double r321924 = sqrt(r321923);
        double r321925 = r321918 - r321921;
        double r321926 = r321924 * r321925;
        double r321927 = r321917 * r321926;
        return r321927;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))