Average Error: 1.0 → 0.0
Time: 36.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\sqrt{2 - \left(v \cdot 6\right) \cdot v}}}{\left(1 - v \cdot v\right) \cdot \left(3 \cdot \pi\right)}\]
\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}{\sqrt{2 - \left(v \cdot 6\right) \cdot v}}}{\left(1 - v \cdot v\right) \cdot \left(3 \cdot \pi\right)}
double f(double v) {
        double r4719659 = 4.0;
        double r4719660 = 3.0;
        double r4719661 = atan2(1.0, 0.0);
        double r4719662 = r4719660 * r4719661;
        double r4719663 = 1.0;
        double r4719664 = v;
        double r4719665 = r4719664 * r4719664;
        double r4719666 = r4719663 - r4719665;
        double r4719667 = r4719662 * r4719666;
        double r4719668 = 2.0;
        double r4719669 = 6.0;
        double r4719670 = r4719669 * r4719665;
        double r4719671 = r4719668 - r4719670;
        double r4719672 = sqrt(r4719671);
        double r4719673 = r4719667 * r4719672;
        double r4719674 = r4719659 / r4719673;
        return r4719674;
}
↓
double f(double v) {
        double r4719675 = 4.0;
        double r4719676 = 2.0;
        double r4719677 = v;
        double r4719678 = 6.0;
        double r4719679 = r4719677 * r4719678;
        double r4719680 = r4719679 * r4719677;
        double r4719681 = r4719676 - r4719680;
        double r4719682 = sqrt(r4719681);
        double r4719683 = r4719675 / r4719682;
        double r4719684 = 1.0;
        double r4719685 = r4719677 * r4719677;
        double r4719686 = r4719684 - r4719685;
        double r4719687 = 3.0;
        double r4719688 = atan2(1.0, 0.0);
        double r4719689 = r4719687 * r4719688;
        double r4719690 = r4719686 * r4719689;
        double r4719691 = r4719683 / r4719690;
        return r4719691;
}

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 *-un-lft-identity1.0

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

  4. Applied times-frac0.0

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

  6. Applied associate-*l/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +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)))))))