Average Error: 1.0 → 0.0
Time: 16.9s
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 r231277 = 4.0;
        double r231278 = 3.0;
        double r231279 = atan2(1.0, 0.0);
        double r231280 = r231278 * r231279;
        double r231281 = 1.0;
        double r231282 = v;
        double r231283 = r231282 * r231282;
        double r231284 = r231281 - r231283;
        double r231285 = r231280 * r231284;
        double r231286 = 2.0;
        double r231287 = 6.0;
        double r231288 = r231287 * r231283;
        double r231289 = r231286 - r231288;
        double r231290 = sqrt(r231289);
        double r231291 = r231285 * r231290;
        double r231292 = r231277 / r231291;
        return r231292;
}
↓
double f(double v) {
        double r231293 = 1.0;
        double r231294 = 3.0;
        double r231295 = atan2(1.0, 0.0);
        double r231296 = r231294 * r231295;
        double r231297 = 1.0;
        double r231298 = v;
        double r231299 = r231298 * r231298;
        double r231300 = r231297 - r231299;
        double r231301 = r231296 * r231300;
        double r231302 = r231293 / r231301;
        double r231303 = 4.0;
        double r231304 = 2.0;
        double r231305 = 6.0;
        double r231306 = r231305 * r231299;
        double r231307 = r231304 - r231306;
        double r231308 = sqrt(r231307);
        double r231309 = r231303 / r231308;
        double r231310 = r231302 * r231309;
        return r231310;
}

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 2019212 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))