Average Error: 1.0 → 0.0
Time: 16.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r5316521 = 4.0;
        double r5316522 = 3.0;
        double r5316523 = atan2(1.0, 0.0);
        double r5316524 = r5316522 * r5316523;
        double r5316525 = 1.0;
        double r5316526 = v;
        double r5316527 = r5316526 * r5316526;
        double r5316528 = r5316525 - r5316527;
        double r5316529 = r5316524 * r5316528;
        double r5316530 = 2.0;
        double r5316531 = 6.0;
        double r5316532 = r5316531 * r5316527;
        double r5316533 = r5316530 - r5316532;
        double r5316534 = sqrt(r5316533);
        double r5316535 = r5316529 * r5316534;
        double r5316536 = r5316521 / r5316535;
        return r5316536;
}
↓
double f(double v) {
        double r5316537 = 1.3333333333333333;
        double r5316538 = atan2(1.0, 0.0);
        double r5316539 = v;
        double r5316540 = r5316539 * r5316539;
        double r5316541 = r5316540 * r5316538;
        double r5316542 = r5316538 - r5316541;
        double r5316543 = r5316537 / r5316542;
        double r5316544 = -6.0;
        double r5316545 = 2.0;
        double r5316546 = fma(r5316544, r5316540, r5316545);
        double r5316547 = sqrt(r5316546);
        double r5316548 = r5316543 / r5316547;
        return r5316548;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))