Average Error: 1.0 → 0.2
Time: 7.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}\]
\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{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}
double f(double v) {
        double r370941 = 4.0;
        double r370942 = 3.0;
        double r370943 = atan2(1.0, 0.0);
        double r370944 = r370942 * r370943;
        double r370945 = 1.0;
        double r370946 = v;
        double r370947 = r370946 * r370946;
        double r370948 = r370945 - r370947;
        double r370949 = r370944 * r370948;
        double r370950 = 2.0;
        double r370951 = 6.0;
        double r370952 = r370951 * r370947;
        double r370953 = r370950 - r370952;
        double r370954 = sqrt(r370953);
        double r370955 = r370949 * r370954;
        double r370956 = r370941 / r370955;
        return r370956;
}
↓
double f(double v) {
        double r370957 = 4.0;
        double r370958 = 3.0;
        double r370959 = 2.0;
        double r370960 = sqrt(r370959);
        double r370961 = r370958 * r370960;
        double r370962 = atan2(1.0, 0.0);
        double r370963 = 9.0;
        double r370964 = v;
        double r370965 = 4.0;
        double r370966 = pow(r370964, r370965);
        double r370967 = r370966 * r370962;
        double r370968 = r370967 / r370960;
        double r370969 = r370963 * r370968;
        double r370970 = 2.0;
        double r370971 = pow(r370964, r370970);
        double r370972 = r370971 * r370962;
        double r370973 = r370972 / r370960;
        double r370974 = 13.5;
        double r370975 = 3.0;
        double r370976 = pow(r370960, r370975);
        double r370977 = r370967 / r370976;
        double r370978 = r370960 * r370972;
        double r370979 = r370958 * r370978;
        double r370980 = fma(r370974, r370977, r370979);
        double r370981 = fma(r370963, r370973, r370980);
        double r370982 = r370969 - r370981;
        double r370983 = fma(r370961, r370962, r370982);
        double r370984 = r370957 / r370983;
        return r370984;
}

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. Taylor expanded around 0 1.1

    \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \left(\sqrt{2} \cdot \pi\right) + 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}}\right) - \left(9 \cdot \frac{{v}^{2} \cdot \pi}{\sqrt{2}} + \left(13.5 \cdot \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}} + 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)}}\]

  3. Simplified0.2

    \[\leadsto \frac{4}{\color{blue}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}}\]

  4. Final simplification0.2

    \[\leadsto \frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))