Average Error: 1.0 → 0.0
Time: 19.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}} \cdot \frac{1}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}\]
\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}}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}} \cdot \frac{1}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}
double f(double v) {
        double r5328813 = 4.0;
        double r5328814 = 3.0;
        double r5328815 = atan2(1.0, 0.0);
        double r5328816 = r5328814 * r5328815;
        double r5328817 = 1.0;
        double r5328818 = v;
        double r5328819 = r5328818 * r5328818;
        double r5328820 = r5328817 - r5328819;
        double r5328821 = r5328816 * r5328820;
        double r5328822 = 2.0;
        double r5328823 = 6.0;
        double r5328824 = r5328823 * r5328819;
        double r5328825 = r5328822 - r5328824;
        double r5328826 = sqrt(r5328825);
        double r5328827 = r5328821 * r5328826;
        double r5328828 = r5328813 / r5328827;
        return r5328828;
}
↓
double f(double v) {
        double r5328829 = 1.3333333333333333;
        double r5328830 = atan2(1.0, 0.0);
        double r5328831 = v;
        double r5328832 = r5328831 * r5328830;
        double r5328833 = r5328831 * r5328832;
        double r5328834 = r5328830 - r5328833;
        double r5328835 = sqrt(r5328834);
        double r5328836 = r5328829 / r5328835;
        double r5328837 = 1.0;
        double r5328838 = r5328837 / r5328835;
        double r5328839 = r5328836 * r5328838;
        double r5328840 = 2.0;
        double r5328841 = r5328831 * r5328831;
        double r5328842 = 6.0;
        double r5328843 = r5328841 * r5328842;
        double r5328844 = r5328840 - r5328843;
        double r5328845 = sqrt(r5328844);
        double r5328846 = r5328839 / r5328845;
        return r5328846;
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))