Average Error: 1.0 → 0.0
Time: 1.4m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{1}{\sqrt{\pi}} \cdot \frac{\frac{4}{3}}{\sqrt{\pi} - \left(v \cdot v\right) \cdot \sqrt{\pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{1}{\sqrt{\pi}} \cdot \frac{\frac{4}{3}}{\sqrt{\pi} - \left(v \cdot v\right) \cdot \sqrt{\pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r13298141 = 4.0;
        double r13298142 = 3.0;
        double r13298143 = atan2(1.0, 0.0);
        double r13298144 = r13298142 * r13298143;
        double r13298145 = 1.0;
        double r13298146 = v;
        double r13298147 = r13298146 * r13298146;
        double r13298148 = r13298145 - r13298147;
        double r13298149 = r13298144 * r13298148;
        double r13298150 = 2.0;
        double r13298151 = 6.0;
        double r13298152 = r13298151 * r13298147;
        double r13298153 = r13298150 - r13298152;
        double r13298154 = sqrt(r13298153);
        double r13298155 = r13298149 * r13298154;
        double r13298156 = r13298141 / r13298155;
        return r13298156;
}
↓
double f(double v) {
        double r13298157 = 1.0;
        double r13298158 = atan2(1.0, 0.0);
        double r13298159 = sqrt(r13298158);
        double r13298160 = r13298157 / r13298159;
        double r13298161 = 1.3333333333333333;
        double r13298162 = v;
        double r13298163 = r13298162 * r13298162;
        double r13298164 = r13298163 * r13298159;
        double r13298165 = r13298159 - r13298164;
        double r13298166 = r13298161 / r13298165;
        double r13298167 = r13298160 * r13298166;
        double r13298168 = -6.0;
        double r13298169 = 2.0;
        double r13298170 = fma(r13298168, r13298163, r13298169);
        double r13298171 = sqrt(r13298170);
        double r13298172 = r13298167 / r13298171;
        return r13298172;
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  6. Applied add-sqr-sqrt0.0

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

  7. Applied distribute-lft-out--0.0

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

  8. Applied *-un-lft-identity0.0

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

  9. Applied times-frac0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))