Average Error: 1.0 → 0.0
Time: 4.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(1 + v \cdot v\right) \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)\]
\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{1}{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(1 + v \cdot v\right) \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)
double f(double v) {
        double r212578 = 4.0;
        double r212579 = 3.0;
        double r212580 = atan2(1.0, 0.0);
        double r212581 = r212579 * r212580;
        double r212582 = 1.0;
        double r212583 = v;
        double r212584 = r212583 * r212583;
        double r212585 = r212582 - r212584;
        double r212586 = r212581 * r212585;
        double r212587 = 2.0;
        double r212588 = 6.0;
        double r212589 = r212588 * r212584;
        double r212590 = r212587 - r212589;
        double r212591 = sqrt(r212590);
        double r212592 = r212586 * r212591;
        double r212593 = r212578 / r212592;
        return r212593;
}
↓
double f(double v) {
        double r212594 = 1.0;
        double r212595 = 3.0;
        double r212596 = atan2(1.0, 0.0);
        double r212597 = r212595 * r212596;
        double r212598 = 1.0;
        double r212599 = r212598 * r212598;
        double r212600 = v;
        double r212601 = r212600 * r212600;
        double r212602 = r212601 * r212601;
        double r212603 = r212599 - r212602;
        double r212604 = r212597 * r212603;
        double r212605 = r212594 / r212604;
        double r212606 = r212598 + r212601;
        double r212607 = 4.0;
        double r212608 = 2.0;
        double r212609 = 6.0;
        double r212610 = r212609 * r212601;
        double r212611 = r212608 - r212610;
        double r212612 = sqrt(r212611);
        double r212613 = r212607 / r212612;
        double r212614 = r212606 * r212613;
        double r212615 = r212605 * r212614;
        return r212615;
}

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 *-un-lft-identity1.0

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

  4. Applied times-frac0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-*r/0.0

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

  8. Applied associate-/r/0.0

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

  9. Applied associate-*l*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))