Average Error: 1.0 → 0.0
Time: 17.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot 1 - {v}^{4}} \cdot \left(1 + v \cdot v\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\frac{4}{3 \cdot \pi}}{1 \cdot 1 - {v}^{4}} \cdot \left(1 + v \cdot v\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r245539 = 4.0;
        double r245540 = 3.0;
        double r245541 = atan2(1.0, 0.0);
        double r245542 = r245540 * r245541;
        double r245543 = 1.0;
        double r245544 = v;
        double r245545 = r245544 * r245544;
        double r245546 = r245543 - r245545;
        double r245547 = r245542 * r245546;
        double r245548 = 2.0;
        double r245549 = 6.0;
        double r245550 = r245549 * r245545;
        double r245551 = r245548 - r245550;
        double r245552 = sqrt(r245551);
        double r245553 = r245547 * r245552;
        double r245554 = r245539 / r245553;
        return r245554;
}
↓
double f(double v) {
        double r245555 = 4.0;
        double r245556 = 3.0;
        double r245557 = atan2(1.0, 0.0);
        double r245558 = r245556 * r245557;
        double r245559 = r245555 / r245558;
        double r245560 = 1.0;
        double r245561 = r245560 * r245560;
        double r245562 = v;
        double r245563 = 4.0;
        double r245564 = pow(r245562, r245563);
        double r245565 = r245561 - r245564;
        double r245566 = r245559 / r245565;
        double r245567 = r245562 * r245562;
        double r245568 = r245560 + r245567;
        double r245569 = r245566 * r245568;
        double r245570 = 2.0;
        double r245571 = 6.0;
        double r245572 = r245571 * r245567;
        double r245573 = r245570 - r245572;
        double r245574 = sqrt(r245573);
        double r245575 = r245569 / r245574;
        return r245575;
}

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. Using strategy rm

  5. Applied flip--0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied associate-/r/0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))