Average Error: 1.0 → 0.0
Time: 13.4s
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 r119985 = 4.0;
        double r119986 = 3.0;
        double r119987 = atan2(1.0, 0.0);
        double r119988 = r119986 * r119987;
        double r119989 = 1.0;
        double r119990 = v;
        double r119991 = r119990 * r119990;
        double r119992 = r119989 - r119991;
        double r119993 = r119988 * r119992;
        double r119994 = 2.0;
        double r119995 = 6.0;
        double r119996 = r119995 * r119991;
        double r119997 = r119994 - r119996;
        double r119998 = sqrt(r119997);
        double r119999 = r119993 * r119998;
        double r120000 = r119985 / r119999;
        return r120000;
}
↓
double f(double v) {
        double r120001 = 4.0;
        double r120002 = 3.0;
        double r120003 = atan2(1.0, 0.0);
        double r120004 = r120002 * r120003;
        double r120005 = 1.0;
        double r120006 = v;
        double r120007 = r120006 * r120006;
        double r120008 = r120005 - r120007;
        double r120009 = r120004 * r120008;
        double r120010 = r120001 / r120009;
        double r120011 = 2.0;
        double r120012 = 6.0;
        double r120013 = r120012 * r120007;
        double r120014 = r120011 - r120013;
        double r120015 = sqrt(r120014);
        double r120016 = r120010 / r120015;
        return r120016;
}

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 2019306 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))