Average Error: 1.0 → 0.0
Time: 28.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(1 \cdot 1 - v \cdot {v}^{3}\right) \cdot \left(3 \cdot \pi\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)\]
\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{4}{\left(1 \cdot 1 - v \cdot {v}^{3}\right) \cdot \left(3 \cdot \pi\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)
double f(double v) {
        double r162018 = 4.0;
        double r162019 = 3.0;
        double r162020 = atan2(1.0, 0.0);
        double r162021 = r162019 * r162020;
        double r162022 = 1.0;
        double r162023 = v;
        double r162024 = r162023 * r162023;
        double r162025 = r162022 - r162024;
        double r162026 = r162021 * r162025;
        double r162027 = 2.0;
        double r162028 = 6.0;
        double r162029 = r162028 * r162024;
        double r162030 = r162027 - r162029;
        double r162031 = sqrt(r162030);
        double r162032 = r162026 * r162031;
        double r162033 = r162018 / r162032;
        return r162033;
}
↓
double f(double v) {
        double r162034 = 4.0;
        double r162035 = 1.0;
        double r162036 = r162035 * r162035;
        double r162037 = v;
        double r162038 = 3.0;
        double r162039 = pow(r162037, r162038);
        double r162040 = r162037 * r162039;
        double r162041 = r162036 - r162040;
        double r162042 = 3.0;
        double r162043 = atan2(1.0, 0.0);
        double r162044 = r162042 * r162043;
        double r162045 = r162041 * r162044;
        double r162046 = r162034 / r162045;
        double r162047 = 2.0;
        double r162048 = 6.0;
        double r162049 = r162037 * r162037;
        double r162050 = r162048 * r162049;
        double r162051 = r162047 - r162050;
        double r162052 = sqrt(r162051);
        double r162053 = r162046 / r162052;
        double r162054 = r162035 + r162049;
        double r162055 = r162053 * r162054;
        return r162055;
}

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 flip--1.0

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))