Average Error: 1.0 → 0.0
Time: 16.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\sqrt[3]{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\left(\mathsf{fma}\left(v \cdot v, -3, 3\right) \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 3\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right)\right) \cdot \mathsf{fma}\left(-6, v \cdot v, 2\right)\right)} \cdot \pi}\]
\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{4}{\sqrt[3]{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \left(\left(\mathsf{fma}\left(v \cdot v, -3, 3\right) \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 3\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right)\right) \cdot \mathsf{fma}\left(-6, v \cdot v, 2\right)\right)} \cdot \pi}
double f(double v) {
        double r3571140 = 4.0;
        double r3571141 = 3.0;
        double r3571142 = atan2(1.0, 0.0);
        double r3571143 = r3571141 * r3571142;
        double r3571144 = 1.0;
        double r3571145 = v;
        double r3571146 = r3571145 * r3571145;
        double r3571147 = r3571144 - r3571146;
        double r3571148 = r3571143 * r3571147;
        double r3571149 = 2.0;
        double r3571150 = 6.0;
        double r3571151 = r3571150 * r3571146;
        double r3571152 = r3571149 - r3571151;
        double r3571153 = sqrt(r3571152);
        double r3571154 = r3571148 * r3571153;
        double r3571155 = r3571140 / r3571154;
        return r3571155;
}
↓
double f(double v) {
        double r3571156 = 4.0;
        double r3571157 = -6.0;
        double r3571158 = v;
        double r3571159 = r3571158 * r3571158;
        double r3571160 = 2.0;
        double r3571161 = fma(r3571157, r3571159, r3571160);
        double r3571162 = sqrt(r3571161);
        double r3571163 = -3.0;
        double r3571164 = 3.0;
        double r3571165 = fma(r3571159, r3571163, r3571164);
        double r3571166 = r3571165 * r3571165;
        double r3571167 = r3571165 * r3571166;
        double r3571168 = r3571167 * r3571161;
        double r3571169 = r3571162 * r3571168;
        double r3571170 = cbrt(r3571169);
        double r3571171 = atan2(1.0, 0.0);
        double r3571172 = r3571170 * r3571171;
        double r3571173 = r3571156 / r3571172;
        return r3571173;
}

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-unprod0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))