Average Error: 1.0 → 0.0
Time: 21.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
\left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot \left(1 \cdot 1\right) - \left(v \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r5115713 = 4.0;
        double r5115714 = 3.0;
        double r5115715 = atan2(1.0, 0.0);
        double r5115716 = r5115714 * r5115715;
        double r5115717 = 1.0;
        double r5115718 = v;
        double r5115719 = r5115718 * r5115718;
        double r5115720 = r5115717 - r5115719;
        double r5115721 = r5115716 * r5115720;
        double r5115722 = 2.0;
        double r5115723 = 6.0;
        double r5115724 = r5115723 * r5115719;
        double r5115725 = r5115722 - r5115724;
        double r5115726 = sqrt(r5115725);
        double r5115727 = r5115721 * r5115726;
        double r5115728 = r5115713 / r5115727;
        return r5115728;
}
↓
double f(double v) {
        double r5115729 = 1.0;
        double r5115730 = r5115729 * r5115729;
        double r5115731 = v;
        double r5115732 = r5115731 * r5115731;
        double r5115733 = r5115732 * r5115729;
        double r5115734 = r5115732 * r5115732;
        double r5115735 = r5115733 + r5115734;
        double r5115736 = r5115730 + r5115735;
        double r5115737 = 4.0;
        double r5115738 = 3.0;
        double r5115739 = atan2(1.0, 0.0);
        double r5115740 = r5115738 * r5115739;
        double r5115741 = r5115737 / r5115740;
        double r5115742 = r5115729 * r5115730;
        double r5115743 = r5115732 * r5115734;
        double r5115744 = r5115742 - r5115743;
        double r5115745 = r5115741 / r5115744;
        double r5115746 = 2.0;
        double r5115747 = 6.0;
        double r5115748 = r5115747 * r5115732;
        double r5115749 = r5115746 - r5115748;
        double r5115750 = sqrt(r5115749);
        double r5115751 = r5115745 / r5115750;
        double r5115752 = r5115736 * r5115751;
        return r5115752;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip3--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))