Average Error: 1.0 → 0.0
Time: 14.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \frac{\sqrt[3]{\frac{4}{3}}}{\pi - \pi \cdot \left(v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \frac{\sqrt[3]{\frac{4}{3}}}{\pi - \pi \cdot \left(v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r5470185 = 4.0;
        double r5470186 = 3.0;
        double r5470187 = atan2(1.0, 0.0);
        double r5470188 = r5470186 * r5470187;
        double r5470189 = 1.0;
        double r5470190 = v;
        double r5470191 = r5470190 * r5470190;
        double r5470192 = r5470189 - r5470191;
        double r5470193 = r5470188 * r5470192;
        double r5470194 = 2.0;
        double r5470195 = 6.0;
        double r5470196 = r5470195 * r5470191;
        double r5470197 = r5470194 - r5470196;
        double r5470198 = sqrt(r5470197);
        double r5470199 = r5470193 * r5470198;
        double r5470200 = r5470185 / r5470199;
        return r5470200;
}
↓
double f(double v) {
        double r5470201 = 1.3333333333333333;
        double r5470202 = cbrt(r5470201);
        double r5470203 = r5470202 * r5470202;
        double r5470204 = atan2(1.0, 0.0);
        double r5470205 = v;
        double r5470206 = r5470205 * r5470205;
        double r5470207 = r5470204 * r5470206;
        double r5470208 = r5470204 - r5470207;
        double r5470209 = r5470202 / r5470208;
        double r5470210 = r5470203 * r5470209;
        double r5470211 = -6.0;
        double r5470212 = 2.0;
        double r5470213 = fma(r5470211, r5470206, r5470212);
        double r5470214 = sqrt(r5470213);
        double r5470215 = r5470210 / r5470214;
        return r5470215;
}

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

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied add-cube-cbrt0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))