Average Error: 1.0 → 0.0
Time: 3.2m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}}{\pi}}{\frac{\sqrt{2 + \left(v \cdot v\right) \cdot -6} - \sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(v \cdot v\right)}{\sqrt[3]{\frac{4}{3}}}}\]
\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{\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}}{\pi}}{\frac{\sqrt{2 + \left(v \cdot v\right) \cdot -6} - \sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(v \cdot v\right)}{\sqrt[3]{\frac{4}{3}}}}
double f(double v) {
        double r48237191 = 4.0;
        double r48237192 = 3.0;
        double r48237193 = atan2(1.0, 0.0);
        double r48237194 = r48237192 * r48237193;
        double r48237195 = 1.0;
        double r48237196 = v;
        double r48237197 = r48237196 * r48237196;
        double r48237198 = r48237195 - r48237197;
        double r48237199 = r48237194 * r48237198;
        double r48237200 = 2.0;
        double r48237201 = 6.0;
        double r48237202 = r48237201 * r48237197;
        double r48237203 = r48237200 - r48237202;
        double r48237204 = sqrt(r48237203);
        double r48237205 = r48237199 * r48237204;
        double r48237206 = r48237191 / r48237205;
        return r48237206;
}
↓
double f(double v) {
        double r48237207 = 1.3333333333333333;
        double r48237208 = cbrt(r48237207);
        double r48237209 = r48237208 * r48237208;
        double r48237210 = atan2(1.0, 0.0);
        double r48237211 = r48237209 / r48237210;
        double r48237212 = 2.0;
        double r48237213 = v;
        double r48237214 = r48237213 * r48237213;
        double r48237215 = -6.0;
        double r48237216 = r48237214 * r48237215;
        double r48237217 = r48237212 + r48237216;
        double r48237218 = sqrt(r48237217);
        double r48237219 = r48237218 * r48237214;
        double r48237220 = r48237218 - r48237219;
        double r48237221 = r48237220 / r48237208;
        double r48237222 = r48237211 / r48237221;
        return r48237222;
}

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

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

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

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

  5. Applied distribute-rgt-out--0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied times-frac0.0

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

  8. Applied associate-/l*0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))