Average Error: 1.0 → 0.0
Time: 4.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r205431 = 4.0;
        double r205432 = 3.0;
        double r205433 = atan2(1.0, 0.0);
        double r205434 = r205432 * r205433;
        double r205435 = 1.0;
        double r205436 = v;
        double r205437 = r205436 * r205436;
        double r205438 = r205435 - r205437;
        double r205439 = r205434 * r205438;
        double r205440 = 2.0;
        double r205441 = 6.0;
        double r205442 = r205441 * r205437;
        double r205443 = r205440 - r205442;
        double r205444 = sqrt(r205443);
        double r205445 = r205439 * r205444;
        double r205446 = r205431 / r205445;
        return r205446;
}
↓
double f(double v) {
        double r205447 = 4.0;
        double r205448 = 3.0;
        double r205449 = atan2(1.0, 0.0);
        double r205450 = r205448 * r205449;
        double r205451 = 1.0;
        double r205452 = v;
        double r205453 = r205452 * r205452;
        double r205454 = r205451 - r205453;
        double r205455 = r205450 * r205454;
        double r205456 = r205447 / r205455;
        double r205457 = 2.0;
        double r205458 = 6.0;
        double r205459 = r205458 * r205453;
        double r205460 = r205457 - r205459;
        double r205461 = sqrt(r205460);
        double r205462 = r205456 / r205461;
        return r205462;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))