Average Error: 1.0 → 0.0
Time: 15.0s
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 r3645927 = 4.0;
        double r3645928 = 3.0;
        double r3645929 = atan2(1.0, 0.0);
        double r3645930 = r3645928 * r3645929;
        double r3645931 = 1.0;
        double r3645932 = v;
        double r3645933 = r3645932 * r3645932;
        double r3645934 = r3645931 - r3645933;
        double r3645935 = r3645930 * r3645934;
        double r3645936 = 2.0;
        double r3645937 = 6.0;
        double r3645938 = r3645937 * r3645933;
        double r3645939 = r3645936 - r3645938;
        double r3645940 = sqrt(r3645939);
        double r3645941 = r3645935 * r3645940;
        double r3645942 = r3645927 / r3645941;
        return r3645942;
}
↓
double f(double v) {
        double r3645943 = 1.3333333333333333;
        double r3645944 = atan2(1.0, 0.0);
        double r3645945 = r3645943 / r3645944;
        double r3645946 = 1.0;
        double r3645947 = v;
        double r3645948 = r3645947 * r3645947;
        double r3645949 = r3645946 - r3645948;
        double r3645950 = r3645945 / r3645949;
        double r3645951 = -6.0;
        double r3645952 = 2.0;
        double r3645953 = fma(r3645951, r3645948, r3645952);
        double r3645954 = sqrt(r3645953);
        double r3645955 = r3645950 / r3645954;
        return r3645955;
}

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)))))))