Average Error: 1.0 → 0.0
Time: 3.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r205248 = 4.0;
        double r205249 = 3.0;
        double r205250 = atan2(1.0, 0.0);
        double r205251 = r205249 * r205250;
        double r205252 = 1.0;
        double r205253 = v;
        double r205254 = r205253 * r205253;
        double r205255 = r205252 - r205254;
        double r205256 = r205251 * r205255;
        double r205257 = 2.0;
        double r205258 = 6.0;
        double r205259 = r205258 * r205254;
        double r205260 = r205257 - r205259;
        double r205261 = sqrt(r205260);
        double r205262 = r205256 * r205261;
        double r205263 = r205248 / r205262;
        return r205263;
}
↓
double f(double v) {
        double r205264 = 1.0;
        double r205265 = 3.0;
        double r205266 = atan2(1.0, 0.0);
        double r205267 = r205265 * r205266;
        double r205268 = 1.0;
        double r205269 = v;
        double r205270 = r205269 * r205269;
        double r205271 = r205268 - r205270;
        double r205272 = r205267 * r205271;
        double r205273 = r205264 / r205272;
        double r205274 = 4.0;
        double r205275 = 2.0;
        double r205276 = 6.0;
        double r205277 = r205276 * r205270;
        double r205278 = r205275 - r205277;
        double r205279 = sqrt(r205278);
        double r205280 = r205274 / r205279;
        double r205281 = r205273 * r205280;
        return r205281;
}

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 *-un-lft-identity1.0

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))