Average Error: 1.0 → 0.0
Time: 1.2m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\pi} \cdot \frac{\frac{4}{3}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} - \sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(v \cdot v\right)}\]
\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{1}{\pi} \cdot \frac{\frac{4}{3}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} - \sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(v \cdot v\right)}
double f(double v) {
        double r33893054 = 4.0;
        double r33893055 = 3.0;
        double r33893056 = atan2(1.0, 0.0);
        double r33893057 = r33893055 * r33893056;
        double r33893058 = 1.0;
        double r33893059 = v;
        double r33893060 = r33893059 * r33893059;
        double r33893061 = r33893058 - r33893060;
        double r33893062 = r33893057 * r33893061;
        double r33893063 = 2.0;
        double r33893064 = 6.0;
        double r33893065 = r33893064 * r33893060;
        double r33893066 = r33893063 - r33893065;
        double r33893067 = sqrt(r33893066);
        double r33893068 = r33893062 * r33893067;
        double r33893069 = r33893054 / r33893068;
        return r33893069;
}
↓
double f(double v) {
        double r33893070 = 1.0;
        double r33893071 = atan2(1.0, 0.0);
        double r33893072 = r33893070 / r33893071;
        double r33893073 = 1.3333333333333333;
        double r33893074 = -6.0;
        double r33893075 = v;
        double r33893076 = r33893075 * r33893075;
        double r33893077 = r33893074 * r33893076;
        double r33893078 = 2.0;
        double r33893079 = r33893077 + r33893078;
        double r33893080 = sqrt(r33893079);
        double r33893081 = r33893080 * r33893076;
        double r33893082 = r33893080 - r33893081;
        double r33893083 = r33893073 / r33893082;
        double r33893084 = r33893072 * r33893083;
        return r33893084;
}

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

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

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

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

  5. Applied sqrt-prod0.0

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

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

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

  7. Applied distribute-rgt-out--0.0

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

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

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

  9. Applied times-frac0.0

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

  10. Applied times-frac0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019104 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))