Average Error: 0.0 → 0.0
Time: 31.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]
\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(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)
double f(double v) {
        double r14237114 = 2.0;
        double r14237115 = sqrt(r14237114);
        double r14237116 = 4.0;
        double r14237117 = r14237115 / r14237116;
        double r14237118 = 1.0;
        double r14237119 = 3.0;
        double r14237120 = v;
        double r14237121 = r14237120 * r14237120;
        double r14237122 = r14237119 * r14237121;
        double r14237123 = r14237118 - r14237122;
        double r14237124 = sqrt(r14237123);
        double r14237125 = r14237117 * r14237124;
        double r14237126 = r14237118 - r14237121;
        double r14237127 = r14237125 * r14237126;
        return r14237127;
}
↓
double f(double v) {
        double r14237128 = 1.0;
        double r14237129 = v;
        double r14237130 = r14237129 * r14237129;
        double r14237131 = r14237128 - r14237130;
        double r14237132 = 3.0;
        double r14237133 = r14237130 * r14237132;
        double r14237134 = r14237128 - r14237133;
        double r14237135 = sqrt(r14237134);
        double r14237136 = sqrt(r14237135);
        double r14237137 = r14237136 * r14237136;
        double r14237138 = 2.0;
        double r14237139 = sqrt(r14237138);
        double r14237140 = 4.0;
        double r14237141 = r14237139 / r14237140;
        double r14237142 = r14237137 * r14237141;
        double r14237143 = r14237131 * r14237142;
        return r14237143;
}

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-sqr-sqrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))