Average Error: 1.0 → 0.0
Time: 16.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot 1 - {v}^{4}} \cdot \left(1 + v \cdot v\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{4}{3 \cdot \pi}}{1 \cdot 1 - {v}^{4}} \cdot \left(1 + v \cdot v\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r254002 = 4.0;
        double r254003 = 3.0;
        double r254004 = atan2(1.0, 0.0);
        double r254005 = r254003 * r254004;
        double r254006 = 1.0;
        double r254007 = v;
        double r254008 = r254007 * r254007;
        double r254009 = r254006 - r254008;
        double r254010 = r254005 * r254009;
        double r254011 = 2.0;
        double r254012 = 6.0;
        double r254013 = r254012 * r254008;
        double r254014 = r254011 - r254013;
        double r254015 = sqrt(r254014);
        double r254016 = r254010 * r254015;
        double r254017 = r254002 / r254016;
        return r254017;
}
↓
double f(double v) {
        double r254018 = 4.0;
        double r254019 = 3.0;
        double r254020 = atan2(1.0, 0.0);
        double r254021 = r254019 * r254020;
        double r254022 = r254018 / r254021;
        double r254023 = 1.0;
        double r254024 = r254023 * r254023;
        double r254025 = v;
        double r254026 = 4.0;
        double r254027 = pow(r254025, r254026);
        double r254028 = r254024 - r254027;
        double r254029 = r254022 / r254028;
        double r254030 = r254025 * r254025;
        double r254031 = r254023 + r254030;
        double r254032 = r254029 * r254031;
        double r254033 = 2.0;
        double r254034 = 6.0;
        double r254035 = r254034 * r254030;
        double r254036 = r254033 - r254035;
        double r254037 = sqrt(r254036);
        double r254038 = r254032 / r254037;
        return r254038;
}

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 associate-/r*0.0

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

  5. Applied flip--0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied associate-/r/0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))