Average Error: 1.0 → 0.0
Time: 12.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r5779123 = 4.0;
        double r5779124 = 3.0;
        double r5779125 = atan2(1.0, 0.0);
        double r5779126 = r5779124 * r5779125;
        double r5779127 = 1.0;
        double r5779128 = v;
        double r5779129 = r5779128 * r5779128;
        double r5779130 = r5779127 - r5779129;
        double r5779131 = r5779126 * r5779130;
        double r5779132 = 2.0;
        double r5779133 = 6.0;
        double r5779134 = r5779133 * r5779129;
        double r5779135 = r5779132 - r5779134;
        double r5779136 = sqrt(r5779135);
        double r5779137 = r5779131 * r5779136;
        double r5779138 = r5779123 / r5779137;
        return r5779138;
}
↓
double f(double v) {
        double r5779139 = 4.0;
        double r5779140 = atan2(1.0, 0.0);
        double r5779141 = 3.0;
        double r5779142 = r5779140 * r5779141;
        double r5779143 = 1.0;
        double r5779144 = v;
        double r5779145 = r5779144 * r5779144;
        double r5779146 = r5779143 - r5779145;
        double r5779147 = r5779142 * r5779146;
        double r5779148 = r5779139 / r5779147;
        double r5779149 = 2.0;
        double r5779150 = 6.0;
        double r5779151 = r5779150 * r5779145;
        double r5779152 = r5779149 - r5779151;
        double r5779153 = sqrt(r5779152);
        double r5779154 = r5779148 / r5779153;
        return r5779154;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))))))