Average Error: 1.0 → 0.0
Time: 15.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{1 \cdot 4}{1 - v \cdot v}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{3 \cdot \pi}\]
\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{\frac{\frac{1 \cdot 4}{1 - v \cdot v}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{3 \cdot \pi}
double f(double v) {
        double r270131 = 4.0;
        double r270132 = 3.0;
        double r270133 = atan2(1.0, 0.0);
        double r270134 = r270132 * r270133;
        double r270135 = 1.0;
        double r270136 = v;
        double r270137 = r270136 * r270136;
        double r270138 = r270135 - r270137;
        double r270139 = r270134 * r270138;
        double r270140 = 2.0;
        double r270141 = 6.0;
        double r270142 = r270141 * r270137;
        double r270143 = r270140 - r270142;
        double r270144 = sqrt(r270143);
        double r270145 = r270139 * r270144;
        double r270146 = r270131 / r270145;
        return r270146;
}
↓
double f(double v) {
        double r270147 = 1.0;
        double r270148 = 4.0;
        double r270149 = r270147 * r270148;
        double r270150 = 1.0;
        double r270151 = v;
        double r270152 = r270151 * r270151;
        double r270153 = r270150 - r270152;
        double r270154 = r270149 / r270153;
        double r270155 = 2.0;
        double r270156 = 6.0;
        double r270157 = r270156 * r270152;
        double r270158 = r270155 - r270157;
        double r270159 = sqrt(r270158);
        double r270160 = r270154 / r270159;
        double r270161 = 3.0;
        double r270162 = atan2(1.0, 0.0);
        double r270163 = r270161 * r270162;
        double r270164 = r270160 / r270163;
        return r270164;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-/r*0.0

    \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
  4. Using strategy rm

  5. Applied flip--0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied associate-/r/0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))