Average Error: 1.0 → 0.0
Time: 17.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{\frac{4}{\pi}} \cdot \frac{\sqrt{\frac{4}{\pi}}}{3 - \left(v \cdot v\right) \cdot 3}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\sqrt{\frac{4}{\pi}} \cdot \frac{\sqrt{\frac{4}{\pi}}}{3 - \left(v \cdot v\right) \cdot 3}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r5069503 = 4.0;
        double r5069504 = 3.0;
        double r5069505 = atan2(1.0, 0.0);
        double r5069506 = r5069504 * r5069505;
        double r5069507 = 1.0;
        double r5069508 = v;
        double r5069509 = r5069508 * r5069508;
        double r5069510 = r5069507 - r5069509;
        double r5069511 = r5069506 * r5069510;
        double r5069512 = 2.0;
        double r5069513 = 6.0;
        double r5069514 = r5069513 * r5069509;
        double r5069515 = r5069512 - r5069514;
        double r5069516 = sqrt(r5069515);
        double r5069517 = r5069511 * r5069516;
        double r5069518 = r5069503 / r5069517;
        return r5069518;
}
↓
double f(double v) {
        double r5069519 = 4.0;
        double r5069520 = atan2(1.0, 0.0);
        double r5069521 = r5069519 / r5069520;
        double r5069522 = sqrt(r5069521);
        double r5069523 = 3.0;
        double r5069524 = v;
        double r5069525 = r5069524 * r5069524;
        double r5069526 = r5069525 * r5069523;
        double r5069527 = r5069523 - r5069526;
        double r5069528 = r5069522 / r5069527;
        double r5069529 = r5069522 * r5069528;
        double r5069530 = 2.0;
        double r5069531 = 6.0;
        double r5069532 = r5069531 * r5069525;
        double r5069533 = r5069530 - r5069532;
        double r5069534 = sqrt(r5069533);
        double r5069535 = r5069529 / r5069534;
        return r5069535;
}

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. Simplified0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))