Average Error: 1.0 → 0.0
Time: 16.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\sqrt{4}}{3}}{1 - v \cdot v} \cdot \frac{\sqrt{4}}{\pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\frac{\sqrt{4}}{3}}{1 - v \cdot v} \cdot \frac{\sqrt{4}}{\pi}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r206946 = 4.0;
        double r206947 = 3.0;
        double r206948 = atan2(1.0, 0.0);
        double r206949 = r206947 * r206948;
        double r206950 = 1.0;
        double r206951 = v;
        double r206952 = r206951 * r206951;
        double r206953 = r206950 - r206952;
        double r206954 = r206949 * r206953;
        double r206955 = 2.0;
        double r206956 = 6.0;
        double r206957 = r206956 * r206952;
        double r206958 = r206955 - r206957;
        double r206959 = sqrt(r206958);
        double r206960 = r206954 * r206959;
        double r206961 = r206946 / r206960;
        return r206961;
}
↓
double f(double v) {
        double r206962 = 4.0;
        double r206963 = sqrt(r206962);
        double r206964 = 3.0;
        double r206965 = r206963 / r206964;
        double r206966 = 1.0;
        double r206967 = v;
        double r206968 = r206967 * r206967;
        double r206969 = r206966 - r206968;
        double r206970 = r206965 / r206969;
        double r206971 = atan2(1.0, 0.0);
        double r206972 = r206963 / r206971;
        double r206973 = r206970 * r206972;
        double r206974 = 2.0;
        double r206975 = 6.0;
        double r206976 = r206975 * r206968;
        double r206977 = r206974 - r206976;
        double r206978 = sqrt(r206977);
        double r206979 = r206973 / r206978;
        return r206979;
}

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

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

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

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))