Average Error: 1.0 → 0.0
Time: 19.0s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]
\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{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)
double f(double v) {
        double r146122 = 4.0;
        double r146123 = 3.0;
        double r146124 = atan2(1.0, 0.0);
        double r146125 = r146123 * r146124;
        double r146126 = 1.0;
        double r146127 = v;
        double r146128 = r146127 * r146127;
        double r146129 = r146126 - r146128;
        double r146130 = r146125 * r146129;
        double r146131 = 2.0;
        double r146132 = 6.0;
        double r146133 = r146132 * r146128;
        double r146134 = r146131 - r146133;
        double r146135 = sqrt(r146134);
        double r146136 = r146130 * r146135;
        double r146137 = r146122 / r146136;
        return r146137;
}
↓
double f(double v) {
        double r146138 = 4.0;
        double r146139 = 3.0;
        double r146140 = atan2(1.0, 0.0);
        double r146141 = r146139 * r146140;
        double r146142 = r146138 / r146141;
        double r146143 = 1.0;
        double r146144 = 3.0;
        double r146145 = pow(r146143, r146144);
        double r146146 = v;
        double r146147 = 6.0;
        double r146148 = pow(r146146, r146147);
        double r146149 = r146145 - r146148;
        double r146150 = r146142 / r146149;
        double r146151 = 2.0;
        double r146152 = 6.0;
        double r146153 = r146146 * r146146;
        double r146154 = r146152 * r146153;
        double r146155 = r146151 - r146154;
        double r146156 = sqrt(r146155);
        double r146157 = r146150 / r146156;
        double r146158 = r146143 * r146143;
        double r146159 = r146153 * r146153;
        double r146160 = r146143 * r146153;
        double r146161 = r146159 + r146160;
        double r146162 = r146158 + r146161;
        double r146163 = r146157 * r146162;
        return r146163;
}

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 flip3--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))