Average Error: 0.0 → 0.0
Time: 17.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{\frac{1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)}{\left(v \cdot v\right) \cdot 3 + 1}}\right)\]
\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 \left(\frac{\sqrt{2}}{4} \cdot \sqrt{\frac{1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)}{\left(v \cdot v\right) \cdot 3 + 1}}\right)
double f(double v) {
        double r4958128 = 2.0;
        double r4958129 = sqrt(r4958128);
        double r4958130 = 4.0;
        double r4958131 = r4958129 / r4958130;
        double r4958132 = 1.0;
        double r4958133 = 3.0;
        double r4958134 = v;
        double r4958135 = r4958134 * r4958134;
        double r4958136 = r4958133 * r4958135;
        double r4958137 = r4958132 - r4958136;
        double r4958138 = sqrt(r4958137);
        double r4958139 = r4958131 * r4958138;
        double r4958140 = r4958132 - r4958135;
        double r4958141 = r4958139 * r4958140;
        return r4958141;
}
↓
double f(double v) {
        double r4958142 = 1.0;
        double r4958143 = v;
        double r4958144 = r4958143 * r4958143;
        double r4958145 = r4958142 - r4958144;
        double r4958146 = 2.0;
        double r4958147 = sqrt(r4958146);
        double r4958148 = 4.0;
        double r4958149 = r4958147 / r4958148;
        double r4958150 = 3.0;
        double r4958151 = r4958144 * r4958150;
        double r4958152 = r4958151 * r4958151;
        double r4958153 = r4958142 - r4958152;
        double r4958154 = r4958151 + r4958142;
        double r4958155 = r4958153 / r4958154;
        double r4958156 = sqrt(r4958155);
        double r4958157 = r4958149 * r4958156;
        double r4958158 = r4958145 * r4958157;
        return r4958158;
}

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 flip--0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))