Average Error: 1.0 → 0.0
Time: 16.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - {v}^{4}\right)}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)\]
\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(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - {v}^{4}\right)}}{\sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)
double f(double v) {
        double r255080 = 4.0;
        double r255081 = 3.0;
        double r255082 = atan2(1.0, 0.0);
        double r255083 = r255081 * r255082;
        double r255084 = 1.0;
        double r255085 = v;
        double r255086 = r255085 * r255085;
        double r255087 = r255084 - r255086;
        double r255088 = r255083 * r255087;
        double r255089 = 2.0;
        double r255090 = 6.0;
        double r255091 = r255090 * r255086;
        double r255092 = r255089 - r255091;
        double r255093 = sqrt(r255092);
        double r255094 = r255088 * r255093;
        double r255095 = r255080 / r255094;
        return r255095;
}
↓
double f(double v) {
        double r255096 = 4.0;
        double r255097 = 3.0;
        double r255098 = atan2(1.0, 0.0);
        double r255099 = r255097 * r255098;
        double r255100 = 1.0;
        double r255101 = r255100 * r255100;
        double r255102 = v;
        double r255103 = 4.0;
        double r255104 = pow(r255102, r255103);
        double r255105 = r255101 - r255104;
        double r255106 = r255099 * r255105;
        double r255107 = r255096 / r255106;
        double r255108 = 2.0;
        double r255109 = 3.0;
        double r255110 = pow(r255108, r255109);
        double r255111 = 6.0;
        double r255112 = r255102 * r255102;
        double r255113 = r255111 * r255112;
        double r255114 = pow(r255113, r255109);
        double r255115 = r255110 - r255114;
        double r255116 = sqrt(r255115);
        double r255117 = r255107 / r255116;
        double r255118 = r255100 + r255112;
        double r255119 = r255108 * r255108;
        double r255120 = r255113 * r255113;
        double r255121 = r255108 * r255113;
        double r255122 = r255120 + r255121;
        double r255123 = r255119 + r255122;
        double r255124 = sqrt(r255123);
        double r255125 = r255118 * r255124;
        double r255126 = r255117 * r255125;
        return r255126;
}

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

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

  4. Applied sqrt-div1.0

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

  5. Applied flip--1.0

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

  6. Applied associate-*r/1.0

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

  7. Applied frac-times1.0

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

  8. Applied associate-/r/1.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))