Average Error: 1.0 → 0.0
Time: 18.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]
\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)}}
↓
\frac{\frac{\frac{4}{3 \cdot \pi}}{{1}^{3} - {v}^{6}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)
double f(double v) {
        double r169388 = 4.0;
        double r169389 = 3.0;
        double r169390 = atan2(1.0, 0.0);
        double r169391 = r169389 * r169390;
        double r169392 = 1.0;
        double r169393 = v;
        double r169394 = r169393 * r169393;
        double r169395 = r169392 - r169394;
        double r169396 = r169391 * r169395;
        double r169397 = 2.0;
        double r169398 = 6.0;
        double r169399 = r169398 * r169394;
        double r169400 = r169397 - r169399;
        double r169401 = sqrt(r169400);
        double r169402 = r169396 * r169401;
        double r169403 = r169388 / r169402;
        return r169403;
}
↓
double f(double v) {
        double r169404 = 4.0;
        double r169405 = 3.0;
        double r169406 = atan2(1.0, 0.0);
        double r169407 = r169405 * r169406;
        double r169408 = r169404 / r169407;
        double r169409 = 1.0;
        double r169410 = 3.0;
        double r169411 = pow(r169409, r169410);
        double r169412 = v;
        double r169413 = 6.0;
        double r169414 = pow(r169412, r169413);
        double r169415 = r169411 - r169414;
        double r169416 = r169408 / r169415;
        double r169417 = 2.0;
        double r169418 = 6.0;
        double r169419 = r169412 * r169412;
        double r169420 = r169418 * r169419;
        double r169421 = r169417 - r169420;
        double r169422 = sqrt(r169421);
        double r169423 = r169416 / r169422;
        double r169424 = r169409 * r169409;
        double r169425 = r169419 * r169419;
        double r169426 = r169409 * r169419;
        double r169427 = r169425 + r169426;
        double r169428 = r169424 + r169427;
        double r169429 = r169423 * r169428;
        return r169429;
}

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 flip3--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))