Average Error: 1.0 → 0.0
Time: 23.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r160141 = 4.0;
        double r160142 = 3.0;
        double r160143 = atan2(1.0, 0.0);
        double r160144 = r160142 * r160143;
        double r160145 = 1.0;
        double r160146 = v;
        double r160147 = r160146 * r160146;
        double r160148 = r160145 - r160147;
        double r160149 = r160144 * r160148;
        double r160150 = 2.0;
        double r160151 = 6.0;
        double r160152 = r160151 * r160147;
        double r160153 = r160150 - r160152;
        double r160154 = sqrt(r160153);
        double r160155 = r160149 * r160154;
        double r160156 = r160141 / r160155;
        return r160156;
}
↓
double f(double v) {
        double r160157 = 1.0;
        double r160158 = v;
        double r160159 = r160158 * r160158;
        double r160160 = 4.0;
        double r160161 = pow(r160158, r160160);
        double r160162 = fma(r160157, r160159, r160161);
        double r160163 = 3.0;
        double r160164 = pow(r160162, r160163);
        double r160165 = 6.0;
        double r160166 = pow(r160157, r160165);
        double r160167 = r160164 + r160166;
        double r160168 = 4.0;
        double r160169 = pow(r160157, r160163);
        double r160170 = pow(r160158, r160165);
        double r160171 = r160169 - r160170;
        double r160172 = r160168 / r160171;
        double r160173 = 3.0;
        double r160174 = atan2(1.0, 0.0);
        double r160175 = r160173 * r160174;
        double r160176 = r160172 / r160175;
        double r160177 = r160167 * r160176;
        double r160178 = fma(r160158, r160158, r160157);
        double r160179 = r160157 * r160157;
        double r160180 = -r160179;
        double r160181 = fma(r160159, r160178, r160180);
        double r160182 = pow(r160157, r160160);
        double r160183 = fma(r160162, r160181, r160182);
        double r160184 = 2.0;
        double r160185 = 6.0;
        double r160186 = r160185 * r160159;
        double r160187 = r160184 - r160186;
        double r160188 = sqrt(r160187);
        double r160189 = r160183 * r160188;
        double r160190 = r160177 / r160189;
        return r160190;
}

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. 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

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

  9. Applied flip3-+0.0

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

  10. Applied frac-times0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))