Average Error: 1.0 → 0.0
Time: 6.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\frac{\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\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)}}
double f(double v) {
        double r501 = 4.0;
        double r502 = 3.0;
        double r503 = atan2(1.0, 0.0);
        double r504 = r502 * r503;
        double r505 = 1.0;
        double r506 = v;
        double r507 = r506 * r506;
        double r508 = r505 - r507;
        double r509 = r504 * r508;
        double r510 = 2.0;
        double r511 = 6.0;
        double r512 = r511 * r507;
        double r513 = r510 - r512;
        double r514 = sqrt(r513);
        double r515 = r509 * r514;
        double r516 = r501 / r515;
        return r516;
}
↓
double f(double v) {
        double r517 = 4.0;
        double r518 = 3.0;
        double r519 = atan2(1.0, 0.0);
        double r520 = r518 * r519;
        double r521 = 1.0;
        double r522 = 3.0;
        double r523 = pow(r521, r522);
        double r524 = v;
        double r525 = r524 * r524;
        double r526 = pow(r525, r522);
        double r527 = r523 - r526;
        double r528 = r520 * r527;
        double r529 = r521 * r521;
        double r530 = r525 * r525;
        double r531 = r521 * r525;
        double r532 = r530 + r531;
        double r533 = r529 + r532;
        double r534 = r528 / r533;
        double r535 = r517 / r534;
        double r536 = 2.0;
        double r537 = 6.0;
        double r538 = r537 * r525;
        double r539 = r536 - r538;
        double r540 = sqrt(r539);
        double r541 = r535 / r540;
        return r541;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))