Average Error: 1.0 → 0.0
Time: 17.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \left(\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)}\right)}\]
\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)}}
↓
\sqrt[3]{\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \left(\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)\right)}\right)}
double f(double v) {
        double r4774474 = 4.0;
        double r4774475 = 3.0;
        double r4774476 = atan2(1.0, 0.0);
        double r4774477 = r4774475 * r4774476;
        double r4774478 = 1.0;
        double r4774479 = v;
        double r4774480 = r4774479 * r4774479;
        double r4774481 = r4774478 - r4774480;
        double r4774482 = r4774477 * r4774481;
        double r4774483 = 2.0;
        double r4774484 = 6.0;
        double r4774485 = r4774484 * r4774480;
        double r4774486 = r4774483 - r4774485;
        double r4774487 = sqrt(r4774486);
        double r4774488 = r4774482 * r4774487;
        double r4774489 = r4774474 / r4774488;
        return r4774489;
}
↓
double f(double v) {
        double r4774490 = 4.0;
        double r4774491 = 2.0;
        double r4774492 = 6.0;
        double r4774493 = v;
        double r4774494 = r4774493 * r4774493;
        double r4774495 = r4774492 * r4774494;
        double r4774496 = r4774491 - r4774495;
        double r4774497 = sqrt(r4774496);
        double r4774498 = atan2(1.0, 0.0);
        double r4774499 = 3.0;
        double r4774500 = r4774498 * r4774499;
        double r4774501 = 1.0;
        double r4774502 = r4774501 - r4774494;
        double r4774503 = r4774500 * r4774502;
        double r4774504 = r4774497 * r4774503;
        double r4774505 = r4774490 / r4774504;
        double r4774506 = r4774505 * r4774505;
        double r4774507 = r4774505 * r4774506;
        double r4774508 = cbrt(r4774507);
        return r4774508;
}

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-cbrt-cube0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))