Average Error: 1.0 → 0.0
Time: 3.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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 r281270 = 4.0;
        double r281271 = 3.0;
        double r281272 = atan2(1.0, 0.0);
        double r281273 = r281271 * r281272;
        double r281274 = 1.0;
        double r281275 = v;
        double r281276 = r281275 * r281275;
        double r281277 = r281274 - r281276;
        double r281278 = r281273 * r281277;
        double r281279 = 2.0;
        double r281280 = 6.0;
        double r281281 = r281280 * r281276;
        double r281282 = r281279 - r281281;
        double r281283 = sqrt(r281282);
        double r281284 = r281278 * r281283;
        double r281285 = r281270 / r281284;
        return r281285;
}
↓
double f(double v) {
        double r281286 = 1.0;
        double r281287 = 3.0;
        double r281288 = atan2(1.0, 0.0);
        double r281289 = r281287 * r281288;
        double r281290 = 1.0;
        double r281291 = v;
        double r281292 = r281291 * r281291;
        double r281293 = r281290 - r281292;
        double r281294 = r281289 * r281293;
        double r281295 = r281286 / r281294;
        double r281296 = 4.0;
        double r281297 = 2.0;
        double r281298 = 6.0;
        double r281299 = r281298 * r281292;
        double r281300 = r281297 - r281299;
        double r281301 = sqrt(r281300);
        double r281302 = r281296 / r281301;
        double r281303 = r281295 * r281302;
        return r281303;
}

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 2020065 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))