Average Error: 1.0 → 0.2
Time: 58.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{(\left(\frac{\frac{4}{3}}{\pi}\right) \cdot \left((\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(v \cdot v\right))_*\right) + \left(\frac{\frac{4}{3}}{\pi}\right))_*}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]
\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{(\left(\frac{\frac{4}{3}}{\pi}\right) \cdot \left((\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(v \cdot v\right))_*\right) + \left(\frac{\frac{4}{3}}{\pi}\right))_*}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}
double f(double v) {
        double r35229303 = 4.0;
        double r35229304 = 3.0;
        double r35229305 = atan2(1.0, 0.0);
        double r35229306 = r35229304 * r35229305;
        double r35229307 = 1.0;
        double r35229308 = v;
        double r35229309 = r35229308 * r35229308;
        double r35229310 = r35229307 - r35229309;
        double r35229311 = r35229306 * r35229310;
        double r35229312 = 2.0;
        double r35229313 = 6.0;
        double r35229314 = r35229313 * r35229309;
        double r35229315 = r35229312 - r35229314;
        double r35229316 = sqrt(r35229315);
        double r35229317 = r35229311 * r35229316;
        double r35229318 = r35229303 / r35229317;
        return r35229318;
}
↓
double f(double v) {
        double r35229319 = 1.3333333333333333;
        double r35229320 = atan2(1.0, 0.0);
        double r35229321 = r35229319 / r35229320;
        double r35229322 = v;
        double r35229323 = r35229322 * r35229322;
        double r35229324 = fma(r35229323, r35229323, r35229323);
        double r35229325 = fma(r35229321, r35229324, r35229321);
        double r35229326 = -6.0;
        double r35229327 = r35229322 * r35229326;
        double r35229328 = 2.0;
        double r35229329 = fma(r35229327, r35229322, r35229328);
        double r35229330 = sqrt(r35229329);
        double r35229331 = r35229325 / r35229330;
        return r35229331;
}

Error

Bits error versus v

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))