Average Error: 1.0 → 0.0
Time: 15.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{1 \cdot 1 - {v}^{4}} \cdot \left(1 + v \cdot v\right)}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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 r253992 = 4.0;
        double r253993 = 3.0;
        double r253994 = atan2(1.0, 0.0);
        double r253995 = r253993 * r253994;
        double r253996 = 1.0;
        double r253997 = v;
        double r253998 = r253997 * r253997;
        double r253999 = r253996 - r253998;
        double r254000 = r253995 * r253999;
        double r254001 = 2.0;
        double r254002 = 6.0;
        double r254003 = r254002 * r253998;
        double r254004 = r254001 - r254003;
        double r254005 = sqrt(r254004);
        double r254006 = r254000 * r254005;
        double r254007 = r253992 / r254006;
        return r254007;
}
↓
double f(double v) {
        double r254008 = 4.0;
        double r254009 = 3.0;
        double r254010 = atan2(1.0, 0.0);
        double r254011 = r254009 * r254010;
        double r254012 = r254008 / r254011;
        double r254013 = 1.0;
        double r254014 = r254013 * r254013;
        double r254015 = v;
        double r254016 = 4.0;
        double r254017 = pow(r254015, r254016);
        double r254018 = r254014 - r254017;
        double r254019 = r254012 / r254018;
        double r254020 = r254015 * r254015;
        double r254021 = r254013 + r254020;
        double r254022 = r254019 * r254021;
        double r254023 = 2.0;
        double r254024 = 6.0;
        double r254025 = r254024 * r254020;
        double r254026 = r254023 - r254025;
        double r254027 = sqrt(r254026);
        double r254028 = r254022 / r254027;
        return r254028;
}

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)))))))