Average Error: 1.0 → 0.0
Time: 6.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
\frac{\sqrt{4}}{\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 \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r239597 = 4.0;
        double r239598 = 3.0;
        double r239599 = atan2(1.0, 0.0);
        double r239600 = r239598 * r239599;
        double r239601 = 1.0;
        double r239602 = v;
        double r239603 = r239602 * r239602;
        double r239604 = r239601 - r239603;
        double r239605 = r239600 * r239604;
        double r239606 = 2.0;
        double r239607 = 6.0;
        double r239608 = r239607 * r239603;
        double r239609 = r239606 - r239608;
        double r239610 = sqrt(r239609);
        double r239611 = r239605 * r239610;
        double r239612 = r239597 / r239611;
        return r239612;
}
↓
double f(double v) {
        double r239613 = 4.0;
        double r239614 = sqrt(r239613);
        double r239615 = 3.0;
        double r239616 = atan2(1.0, 0.0);
        double r239617 = r239615 * r239616;
        double r239618 = 1.0;
        double r239619 = 3.0;
        double r239620 = pow(r239618, r239619);
        double r239621 = v;
        double r239622 = r239621 * r239621;
        double r239623 = pow(r239622, r239619);
        double r239624 = r239620 - r239623;
        double r239625 = r239617 * r239624;
        double r239626 = r239618 * r239618;
        double r239627 = r239622 * r239622;
        double r239628 = r239618 * r239622;
        double r239629 = r239627 + r239628;
        double r239630 = r239626 + r239629;
        double r239631 = r239625 / r239630;
        double r239632 = r239614 / r239631;
        double r239633 = 2.0;
        double r239634 = 6.0;
        double r239635 = r239634 * r239622;
        double r239636 = r239633 - r239635;
        double r239637 = sqrt(r239636);
        double r239638 = r239614 / r239637;
        double r239639 = r239632 * r239638;
        return r239639;
}

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 add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  6. Applied flip3--0.0

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

  7. Applied associate-*r/0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))