Average Error: 1.0 → 0.0
Time: 14.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r7757426 = 4.0;
        double r7757427 = 3.0;
        double r7757428 = atan2(1.0, 0.0);
        double r7757429 = r7757427 * r7757428;
        double r7757430 = 1.0;
        double r7757431 = v;
        double r7757432 = r7757431 * r7757431;
        double r7757433 = r7757430 - r7757432;
        double r7757434 = r7757429 * r7757433;
        double r7757435 = 2.0;
        double r7757436 = 6.0;
        double r7757437 = r7757436 * r7757432;
        double r7757438 = r7757435 - r7757437;
        double r7757439 = sqrt(r7757438);
        double r7757440 = r7757434 * r7757439;
        double r7757441 = r7757426 / r7757440;
        return r7757441;
}
↓
double f(double v) {
        double r7757442 = 4.0;
        double r7757443 = atan2(1.0, 0.0);
        double r7757444 = 3.0;
        double r7757445 = r7757443 * r7757444;
        double r7757446 = 1.0;
        double r7757447 = v;
        double r7757448 = r7757447 * r7757447;
        double r7757449 = r7757446 - r7757448;
        double r7757450 = r7757445 * r7757449;
        double r7757451 = r7757442 / r7757450;
        double r7757452 = 2.0;
        double r7757453 = 6.0;
        double r7757454 = r7757453 * r7757448;
        double r7757455 = r7757452 - r7757454;
        double r7757456 = sqrt(r7757455);
        double r7757457 = r7757451 / r7757456;
        return r7757457;
}

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

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

Reproduce

herbie shell --seed 2019171 +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)))))))