Average Error: 1.0 → 0.0
Time: 8.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\frac{\left({1}^{3} - {v}^{6}\right) \cdot \left(3 \cdot \pi\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{4}{\frac{\left({1}^{3} - {v}^{6}\right) \cdot \left(3 \cdot \pi\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r293112 = 4.0;
        double r293113 = 3.0;
        double r293114 = atan2(1.0, 0.0);
        double r293115 = r293113 * r293114;
        double r293116 = 1.0;
        double r293117 = v;
        double r293118 = r293117 * r293117;
        double r293119 = r293116 - r293118;
        double r293120 = r293115 * r293119;
        double r293121 = 2.0;
        double r293122 = 6.0;
        double r293123 = r293122 * r293118;
        double r293124 = r293121 - r293123;
        double r293125 = sqrt(r293124);
        double r293126 = r293120 * r293125;
        double r293127 = r293112 / r293126;
        return r293127;
}
↓
double f(double v) {
        double r293128 = 4.0;
        double r293129 = 1.0;
        double r293130 = 3.0;
        double r293131 = pow(r293129, r293130);
        double r293132 = v;
        double r293133 = 6.0;
        double r293134 = pow(r293132, r293133);
        double r293135 = r293131 - r293134;
        double r293136 = 3.0;
        double r293137 = atan2(1.0, 0.0);
        double r293138 = r293136 * r293137;
        double r293139 = r293135 * r293138;
        double r293140 = r293129 * r293129;
        double r293141 = r293132 * r293132;
        double r293142 = r293141 * r293141;
        double r293143 = r293129 * r293141;
        double r293144 = r293142 + r293143;
        double r293145 = r293140 + r293144;
        double r293146 = r293139 / r293145;
        double r293147 = r293128 / r293146;
        double r293148 = 2.0;
        double r293149 = 6.0;
        double r293150 = r293149 * r293141;
        double r293151 = r293148 - r293150;
        double r293152 = sqrt(r293151);
        double r293153 = r293147 / r293152;
        return r293153;
}

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 flip3--0.0

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

  6. Applied associate-*r/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))