Average Error: 1.0 → 0.0
Time: 3.3m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\frac{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}}\]
double f(double v) {
        double r21904112 = 4.0;
        double r21904113 = 3.0;
        double r21904114 = atan2(1.0, 0.0);
        double r21904115 = r21904113 * r21904114;
        double r21904116 = 1.0;
        double r21904117 = v;
        double r21904118 = r21904117 * r21904117;
        double r21904119 = r21904116 - r21904118;
        double r21904120 = r21904115 * r21904119;
        double r21904121 = 2.0;
        double r21904122 = 6.0;
        double r21904123 = r21904122 * r21904118;
        double r21904124 = r21904121 - r21904123;
        double r21904125 = sqrt(r21904124);
        double r21904126 = r21904120 * r21904125;
        double r21904127 = r21904112 / r21904126;
        return r21904127;
}
↓
double f(double v) {
        double r21904128 = 1.0;
        double r21904129 = -6.0;
        double r21904130 = v;
        double r21904131 = r21904129 * r21904130;
        double r21904132 = 2.0;
        double r21904133 = fma(r21904131, r21904130, r21904132);
        double r21904134 = sqrt(r21904133);
        double r21904135 = 1.3333333333333333;
        double r21904136 = atan2(1.0, 0.0);
        double r21904137 = r21904136 * r21904130;
        double r21904138 = r21904130 * r21904137;
        double r21904139 = r21904136 - r21904138;
        double r21904140 = r21904135 / r21904139;
        double r21904141 = r21904134 / r21904140;
        double r21904142 = r21904128 / r21904141;
        return r21904142;
}


\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{1}{\frac{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}}

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]

  5. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}}}\]

  6. Final simplification0.0

    \[\leadsto \frac{1}{\frac{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))