Average Error: 1.0 → 0.0
Time: 24.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{1}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \cdot \frac{4}{3}\]
\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{1}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \cdot \frac{4}{3}
double f(double v) {
        double r6372665 = 4.0;
        double r6372666 = 3.0;
        double r6372667 = atan2(1.0, 0.0);
        double r6372668 = r6372666 * r6372667;
        double r6372669 = 1.0;
        double r6372670 = v;
        double r6372671 = r6372670 * r6372670;
        double r6372672 = r6372669 - r6372671;
        double r6372673 = r6372668 * r6372672;
        double r6372674 = 2.0;
        double r6372675 = 6.0;
        double r6372676 = r6372675 * r6372671;
        double r6372677 = r6372674 - r6372676;
        double r6372678 = sqrt(r6372677);
        double r6372679 = r6372673 * r6372678;
        double r6372680 = r6372665 / r6372679;
        return r6372680;
}
↓
double f(double v) {
        double r6372681 = 1.0;
        double r6372682 = atan2(1.0, 0.0);
        double r6372683 = v;
        double r6372684 = r6372683 * r6372683;
        double r6372685 = r6372684 * r6372682;
        double r6372686 = r6372682 - r6372685;
        double r6372687 = r6372681 / r6372686;
        double r6372688 = -6.0;
        double r6372689 = 2.0;
        double r6372690 = fma(r6372688, r6372684, r6372689);
        double r6372691 = sqrt(r6372690);
        double r6372692 = r6372687 / r6372691;
        double r6372693 = 1.3333333333333333;
        double r6372694 = r6372692 * r6372693;
        return r6372694;
}

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

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

  5. Applied sqrt-prod0.0

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

  6. Applied div-inv0.0

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))