Average Error: 1.0 → 0.0
Time: 22.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{\left(1 - v \cdot v\right) \cdot \pi}}{3}\]
\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}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{\left(1 - v \cdot v\right) \cdot \pi}}{3}
double f(double v) {
        double r202800 = 4.0;
        double r202801 = 3.0;
        double r202802 = atan2(1.0, 0.0);
        double r202803 = r202801 * r202802;
        double r202804 = 1.0;
        double r202805 = v;
        double r202806 = r202805 * r202805;
        double r202807 = r202804 - r202806;
        double r202808 = r202803 * r202807;
        double r202809 = 2.0;
        double r202810 = 6.0;
        double r202811 = r202810 * r202806;
        double r202812 = r202809 - r202811;
        double r202813 = sqrt(r202812);
        double r202814 = r202808 * r202813;
        double r202815 = r202800 / r202814;
        return r202815;
}
↓
double f(double v) {
        double r202816 = 4.0;
        double r202817 = 2.0;
        double r202818 = 6.0;
        double r202819 = v;
        double r202820 = r202819 * r202819;
        double r202821 = r202818 * r202820;
        double r202822 = r202817 - r202821;
        double r202823 = sqrt(r202822);
        double r202824 = r202816 / r202823;
        double r202825 = 1.0;
        double r202826 = r202825 - r202820;
        double r202827 = atan2(1.0, 0.0);
        double r202828 = r202826 * r202827;
        double r202829 = r202824 / r202828;
        double r202830 = 3.0;
        double r202831 = r202829 / r202830;
        return r202831;
}

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 add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  6. Applied associate-/r*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019294 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))