Average Error: 1.0 → 0.0
Time: 3.0m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}\]
\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{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}
double f(double v) {
        double r39432228 = 4.0;
        double r39432229 = 3.0;
        double r39432230 = atan2(1.0, 0.0);
        double r39432231 = r39432229 * r39432230;
        double r39432232 = 1.0;
        double r39432233 = v;
        double r39432234 = r39432233 * r39432233;
        double r39432235 = r39432232 - r39432234;
        double r39432236 = r39432231 * r39432235;
        double r39432237 = 2.0;
        double r39432238 = 6.0;
        double r39432239 = r39432238 * r39432234;
        double r39432240 = r39432237 - r39432239;
        double r39432241 = sqrt(r39432240);
        double r39432242 = r39432236 * r39432241;
        double r39432243 = r39432228 / r39432242;
        return r39432243;
}
↓
double f(double v) {
        double r39432244 = 1.3333333333333333;
        double r39432245 = atan2(1.0, 0.0);
        double r39432246 = r39432244 / r39432245;
        double r39432247 = 1.0;
        double r39432248 = v;
        double r39432249 = r39432248 * r39432248;
        double r39432250 = r39432247 - r39432249;
        double r39432251 = r39432246 / r39432250;
        double r39432252 = 2.0;
        double r39432253 = -6.0;
        double r39432254 = r39432248 * r39432253;
        double r39432255 = r39432248 * r39432254;
        double r39432256 = r39432252 + r39432255;
        double r39432257 = sqrt(r39432256);
        double r39432258 = r39432251 / r39432257;
        return r39432258;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

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

  5. Applied distribute-rgt-out--0.0

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

  6. Applied associate-/r*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019119 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))