Average Error: 1.0 → 0.0
Time: 19.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r2284771 = 4.0;
        double r2284772 = 3.0;
        double r2284773 = atan2(1.0, 0.0);
        double r2284774 = r2284772 * r2284773;
        double r2284775 = 1.0;
        double r2284776 = v;
        double r2284777 = r2284776 * r2284776;
        double r2284778 = r2284775 - r2284777;
        double r2284779 = r2284774 * r2284778;
        double r2284780 = 2.0;
        double r2284781 = 6.0;
        double r2284782 = r2284781 * r2284777;
        double r2284783 = r2284780 - r2284782;
        double r2284784 = sqrt(r2284783);
        double r2284785 = r2284779 * r2284784;
        double r2284786 = r2284771 / r2284785;
        return r2284786;
}
↓
double f(double v) {
        double r2284787 = 1.3333333333333333;
        double r2284788 = atan2(1.0, 0.0);
        double r2284789 = r2284787 / r2284788;
        double r2284790 = 1.0;
        double r2284791 = v;
        double r2284792 = r2284791 * r2284791;
        double r2284793 = r2284790 - r2284792;
        double r2284794 = r2284789 / r2284793;
        double r2284795 = -6.0;
        double r2284796 = 2.0;
        double r2284797 = fma(r2284795, r2284792, r2284796);
        double r2284798 = sqrt(r2284797);
        double r2284799 = r2284794 / r2284798;
        return r2284799;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))