Average Error: 1.0 → 0.0
Time: 22.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}\]
\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({\left(1 \cdot 1\right)}^{3} + {\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right)}^{3}\right) \cdot \frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v + 1\right) - 1 \cdot 1\right) + 1 \cdot {1}^{3}\right)}
double f(double v) {
        double r245560 = 4.0;
        double r245561 = 3.0;
        double r245562 = atan2(1.0, 0.0);
        double r245563 = r245561 * r245562;
        double r245564 = 1.0;
        double r245565 = v;
        double r245566 = r245565 * r245565;
        double r245567 = r245564 - r245566;
        double r245568 = r245563 * r245567;
        double r245569 = 2.0;
        double r245570 = 6.0;
        double r245571 = r245570 * r245566;
        double r245572 = r245569 - r245571;
        double r245573 = sqrt(r245572);
        double r245574 = r245568 * r245573;
        double r245575 = r245560 / r245574;
        return r245575;
}
↓
double f(double v) {
        double r245576 = 1.0;
        double r245577 = r245576 * r245576;
        double r245578 = 3.0;
        double r245579 = pow(r245577, r245578);
        double r245580 = v;
        double r245581 = r245580 * r245580;
        double r245582 = r245581 + r245576;
        double r245583 = r245581 * r245582;
        double r245584 = pow(r245583, r245578);
        double r245585 = r245579 + r245584;
        double r245586 = 4.0;
        double r245587 = 3.0;
        double r245588 = atan2(1.0, 0.0);
        double r245589 = r245587 * r245588;
        double r245590 = r245586 / r245589;
        double r245591 = pow(r245576, r245578);
        double r245592 = 6.0;
        double r245593 = pow(r245580, r245592);
        double r245594 = r245591 - r245593;
        double r245595 = r245590 / r245594;
        double r245596 = r245585 * r245595;
        double r245597 = 2.0;
        double r245598 = 6.0;
        double r245599 = r245598 * r245581;
        double r245600 = r245597 - r245599;
        double r245601 = sqrt(r245600);
        double r245602 = r245583 - r245577;
        double r245603 = r245583 * r245602;
        double r245604 = r245576 * r245591;
        double r245605 = r245603 + r245604;
        double r245606 = r245601 * r245605;
        double r245607 = r245596 / r245606;
        return r245607;
}

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

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

  9. Applied flip3-+0.0

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

  10. Applied frac-times0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))