Average Error: 0.0 → 0.0
Time: 19.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \sqrt[3]{\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right)\right)}
double f(double v) {
        double r3813122 = 2.0;
        double r3813123 = sqrt(r3813122);
        double r3813124 = 4.0;
        double r3813125 = r3813123 / r3813124;
        double r3813126 = 1.0;
        double r3813127 = 3.0;
        double r3813128 = v;
        double r3813129 = r3813128 * r3813128;
        double r3813130 = r3813127 * r3813129;
        double r3813131 = r3813126 - r3813130;
        double r3813132 = sqrt(r3813131);
        double r3813133 = r3813125 * r3813132;
        double r3813134 = r3813126 - r3813129;
        double r3813135 = r3813133 * r3813134;
        return r3813135;
}
↓
double f(double v) {
        double r3813136 = 1.0;
        double r3813137 = v;
        double r3813138 = r3813137 * r3813137;
        double r3813139 = r3813136 - r3813138;
        double r3813140 = 3.0;
        double r3813141 = r3813138 * r3813140;
        double r3813142 = r3813136 - r3813141;
        double r3813143 = sqrt(r3813142);
        double r3813144 = 2.0;
        double r3813145 = sqrt(r3813144);
        double r3813146 = 4.0;
        double r3813147 = r3813145 / r3813146;
        double r3813148 = r3813143 * r3813147;
        double r3813149 = r3813148 * r3813148;
        double r3813150 = r3813148 * r3813149;
        double r3813151 = cbrt(r3813150);
        double r3813152 = r3813139 * r3813151;
        return r3813152;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))