Average Error: 1.0 → 0.0
Time: 17.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}} \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\]
\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{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}} \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}
double f(double v) {
        double r224118 = 4.0;
        double r224119 = 3.0;
        double r224120 = atan2(1.0, 0.0);
        double r224121 = r224119 * r224120;
        double r224122 = 1.0;
        double r224123 = v;
        double r224124 = r224123 * r224123;
        double r224125 = r224122 - r224124;
        double r224126 = r224121 * r224125;
        double r224127 = 2.0;
        double r224128 = 6.0;
        double r224129 = r224128 * r224124;
        double r224130 = r224127 - r224129;
        double r224131 = sqrt(r224130);
        double r224132 = r224126 * r224131;
        double r224133 = r224118 / r224132;
        return r224133;
}
↓
double f(double v) {
        double r224134 = 4.0;
        double r224135 = 3.0;
        double r224136 = atan2(1.0, 0.0);
        double r224137 = r224135 * r224136;
        double r224138 = 1.0;
        double r224139 = v;
        double r224140 = r224139 * r224139;
        double r224141 = r224138 - r224140;
        double r224142 = r224137 * r224141;
        double r224143 = r224134 / r224142;
        double r224144 = 2.0;
        double r224145 = 3.0;
        double r224146 = pow(r224144, r224145);
        double r224147 = 6.0;
        double r224148 = r224147 * r224140;
        double r224149 = pow(r224148, r224145);
        double r224150 = r224146 - r224149;
        double r224151 = sqrt(r224150);
        double r224152 = r224143 / r224151;
        double r224153 = r224144 * r224144;
        double r224154 = r224148 * r224148;
        double r224155 = r224144 * r224148;
        double r224156 = r224154 + r224155;
        double r224157 = r224153 + r224156;
        double r224158 = sqrt(r224157);
        double r224159 = r224152 * r224158;
        return r224159;
}

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

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

  6. Applied sqrt-div0.0

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

  7. Applied associate-/r/0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))