Average Error: 1.0 → 0.0
Time: 6.7s
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 r3549187 = 4.0;
        double r3549188 = 3.0;
        double r3549189 = atan2(1.0, 0.0);
        double r3549190 = r3549188 * r3549189;
        double r3549191 = 1.0;
        double r3549192 = v;
        double r3549193 = r3549192 * r3549192;
        double r3549194 = r3549191 - r3549193;
        double r3549195 = r3549190 * r3549194;
        double r3549196 = 2.0;
        double r3549197 = 6.0;
        double r3549198 = r3549197 * r3549193;
        double r3549199 = r3549196 - r3549198;
        double r3549200 = sqrt(r3549199);
        double r3549201 = r3549195 * r3549200;
        double r3549202 = r3549187 / r3549201;
        return r3549202;
}
↓
double f(double v) {
        double r3549203 = 1.3333333333333333;
        double r3549204 = atan2(1.0, 0.0);
        double r3549205 = r3549203 / r3549204;
        double r3549206 = 1.0;
        double r3549207 = v;
        double r3549208 = r3549207 * r3549207;
        double r3549209 = r3549206 - r3549208;
        double r3549210 = r3549205 / r3549209;
        double r3549211 = -6.0;
        double r3549212 = 2.0;
        double r3549213 = fma(r3549211, r3549208, r3549212);
        double r3549214 = sqrt(r3549213);
        double r3549215 = r3549210 / r3549214;
        return r3549215;
}

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 2019154 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))