Average Error: 1.0 → 0.0
Time: 16.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\sqrt{4}}{3}}{1 - v \cdot v} \cdot \frac{\sqrt{4}}{\pi}}{\sqrt{2 - \left(v \cdot 6\right) \cdot v}}\]
\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{\sqrt{4}}{3}}{1 - v \cdot v} \cdot \frac{\sqrt{4}}{\pi}}{\sqrt{2 - \left(v \cdot 6\right) \cdot v}}
double f(double v) {
        double r153707 = 4.0;
        double r153708 = 3.0;
        double r153709 = atan2(1.0, 0.0);
        double r153710 = r153708 * r153709;
        double r153711 = 1.0;
        double r153712 = v;
        double r153713 = r153712 * r153712;
        double r153714 = r153711 - r153713;
        double r153715 = r153710 * r153714;
        double r153716 = 2.0;
        double r153717 = 6.0;
        double r153718 = r153717 * r153713;
        double r153719 = r153716 - r153718;
        double r153720 = sqrt(r153719);
        double r153721 = r153715 * r153720;
        double r153722 = r153707 / r153721;
        return r153722;
}
↓
double f(double v) {
        double r153723 = 4.0;
        double r153724 = sqrt(r153723);
        double r153725 = 3.0;
        double r153726 = r153724 / r153725;
        double r153727 = 1.0;
        double r153728 = v;
        double r153729 = r153728 * r153728;
        double r153730 = r153727 - r153729;
        double r153731 = r153726 / r153730;
        double r153732 = atan2(1.0, 0.0);
        double r153733 = r153724 / r153732;
        double r153734 = r153731 * r153733;
        double r153735 = 2.0;
        double r153736 = 6.0;
        double r153737 = r153728 * r153736;
        double r153738 = r153737 * r153728;
        double r153739 = r153735 - r153738;
        double r153740 = sqrt(r153739);
        double r153741 = r153734 / r153740;
        return r153741;
}

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. Simplified0.0

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

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

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))