Average Error: 1.0 → 0.0
Time: 21.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\left(1 \cdot 1 + \left(1 \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3}}{\left(1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)\right) \cdot \pi}}{\sqrt{2 - v \cdot \left(6 \cdot v\right)}}\]
\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)}}
↓
\left(1 \cdot 1 + \left(1 \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{\frac{4}{3}}{\left(1 \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)\right) \cdot \pi}}{\sqrt{2 - v \cdot \left(6 \cdot v\right)}}
double f(double v) {
        double r6872012 = 4.0;
        double r6872013 = 3.0;
        double r6872014 = atan2(1.0, 0.0);
        double r6872015 = r6872013 * r6872014;
        double r6872016 = 1.0;
        double r6872017 = v;
        double r6872018 = r6872017 * r6872017;
        double r6872019 = r6872016 - r6872018;
        double r6872020 = r6872015 * r6872019;
        double r6872021 = 2.0;
        double r6872022 = 6.0;
        double r6872023 = r6872022 * r6872018;
        double r6872024 = r6872021 - r6872023;
        double r6872025 = sqrt(r6872024);
        double r6872026 = r6872020 * r6872025;
        double r6872027 = r6872012 / r6872026;
        return r6872027;
}
↓
double f(double v) {
        double r6872028 = 1.0;
        double r6872029 = r6872028 * r6872028;
        double r6872030 = v;
        double r6872031 = r6872030 * r6872030;
        double r6872032 = r6872028 * r6872031;
        double r6872033 = r6872031 * r6872031;
        double r6872034 = r6872032 + r6872033;
        double r6872035 = r6872029 + r6872034;
        double r6872036 = 4.0;
        double r6872037 = 3.0;
        double r6872038 = r6872036 / r6872037;
        double r6872039 = r6872028 * r6872029;
        double r6872040 = r6872031 * r6872030;
        double r6872041 = r6872040 * r6872040;
        double r6872042 = r6872039 - r6872041;
        double r6872043 = atan2(1.0, 0.0);
        double r6872044 = r6872042 * r6872043;
        double r6872045 = r6872038 / r6872044;
        double r6872046 = 2.0;
        double r6872047 = 6.0;
        double r6872048 = r6872047 * r6872030;
        double r6872049 = r6872030 * r6872048;
        double r6872050 = r6872046 - r6872049;
        double r6872051 = sqrt(r6872050);
        double r6872052 = r6872045 / r6872051;
        double r6872053 = r6872035 * r6872052;
        return r6872053;
}

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 flip3--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +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)))))))