Average Error: 1.0 → 0.0
Time: 19.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\frac{3 \cdot \left(\pi \cdot \left({1}^{3} - {v}^{6}\right)\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
\frac{\frac{4}{\frac{3 \cdot \left(\pi \cdot \left({1}^{3} - {v}^{6}\right)\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r242550 = 4.0;
        double r242551 = 3.0;
        double r242552 = atan2(1.0, 0.0);
        double r242553 = r242551 * r242552;
        double r242554 = 1.0;
        double r242555 = v;
        double r242556 = r242555 * r242555;
        double r242557 = r242554 - r242556;
        double r242558 = r242553 * r242557;
        double r242559 = 2.0;
        double r242560 = 6.0;
        double r242561 = r242560 * r242556;
        double r242562 = r242559 - r242561;
        double r242563 = sqrt(r242562);
        double r242564 = r242558 * r242563;
        double r242565 = r242550 / r242564;
        return r242565;
}
↓
double f(double v) {
        double r242566 = 4.0;
        double r242567 = 3.0;
        double r242568 = atan2(1.0, 0.0);
        double r242569 = 1.0;
        double r242570 = 3.0;
        double r242571 = pow(r242569, r242570);
        double r242572 = v;
        double r242573 = 6.0;
        double r242574 = pow(r242572, r242573);
        double r242575 = r242571 - r242574;
        double r242576 = r242568 * r242575;
        double r242577 = r242567 * r242576;
        double r242578 = r242569 * r242569;
        double r242579 = r242572 * r242572;
        double r242580 = r242579 * r242579;
        double r242581 = r242569 * r242579;
        double r242582 = r242580 + r242581;
        double r242583 = r242578 + r242582;
        double r242584 = r242577 / r242583;
        double r242585 = r242566 / r242584;
        double r242586 = 2.0;
        double r242587 = 6.0;
        double r242588 = r242587 * r242579;
        double r242589 = r242586 - r242588;
        double r242590 = sqrt(r242589);
        double r242591 = r242585 / r242590;
        return r242591;
}

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 associate-/r*0.0

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

  5. Applied flip3--0.0

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

  6. Applied associate-*r/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))