Average Error: 0.0 → 0.0
Time: 24.0s
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 r12350108 = 2.0;
        double r12350109 = sqrt(r12350108);
        double r12350110 = 4.0;
        double r12350111 = r12350109 / r12350110;
        double r12350112 = 1.0;
        double r12350113 = 3.0;
        double r12350114 = v;
        double r12350115 = r12350114 * r12350114;
        double r12350116 = r12350113 * r12350115;
        double r12350117 = r12350112 - r12350116;
        double r12350118 = sqrt(r12350117);
        double r12350119 = r12350111 * r12350118;
        double r12350120 = r12350112 - r12350115;
        double r12350121 = r12350119 * r12350120;
        return r12350121;
}
↓
double f(double v) {
        double r12350122 = 1.0;
        double r12350123 = v;
        double r12350124 = r12350123 * r12350123;
        double r12350125 = r12350122 - r12350124;
        double r12350126 = 3.0;
        double r12350127 = r12350124 * r12350126;
        double r12350128 = r12350122 - r12350127;
        double r12350129 = sqrt(r12350128);
        double r12350130 = 2.0;
        double r12350131 = sqrt(r12350130);
        double r12350132 = 4.0;
        double r12350133 = r12350131 / r12350132;
        double r12350134 = r12350129 * r12350133;
        double r12350135 = r12350134 * r12350134;
        double r12350136 = r12350134 * r12350135;
        double r12350137 = cbrt(r12350136);
        double r12350138 = r12350125 * r12350137;
        return r12350138;
}

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 2019174 +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))))