Average Error: 1.0 → 0.0
Time: 30.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(v \cdot \pi\right)}}{\sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}}\]
\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 - v \cdot \left(v \cdot \pi\right)}}{\sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}}
double f(double v) {
        double r27049321 = 4.0;
        double r27049322 = 3.0;
        double r27049323 = atan2(1.0, 0.0);
        double r27049324 = r27049322 * r27049323;
        double r27049325 = 1.0;
        double r27049326 = v;
        double r27049327 = r27049326 * r27049326;
        double r27049328 = r27049325 - r27049327;
        double r27049329 = r27049324 * r27049328;
        double r27049330 = 2.0;
        double r27049331 = 6.0;
        double r27049332 = r27049331 * r27049327;
        double r27049333 = r27049330 - r27049332;
        double r27049334 = sqrt(r27049333);
        double r27049335 = r27049329 * r27049334;
        double r27049336 = r27049321 / r27049335;
        return r27049336;
}
↓
double f(double v) {
        double r27049337 = 1.3333333333333333;
        double r27049338 = atan2(1.0, 0.0);
        double r27049339 = v;
        double r27049340 = r27049339 * r27049338;
        double r27049341 = r27049339 * r27049340;
        double r27049342 = r27049338 - r27049341;
        double r27049343 = r27049337 / r27049342;
        double r27049344 = -6.0;
        double r27049345 = r27049344 * r27049339;
        double r27049346 = 2.0;
        double r27049347 = fma(r27049345, r27049339, r27049346);
        double r27049348 = sqrt(r27049347);
        double r27049349 = r27049343 / r27049348;
        return r27049349;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))