Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Average Error: 9.6 → 0.4

Time: 14.9s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

↓

\[\left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2 + \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r2248 = x;
        double r2249 = y;
        double r2250 = log(r2249);
        double r2251 = r2248 * r2250;
        double r2252 = z;
        double r2253 = 1.0;
        double r2254 = r2253 - r2249;
        double r2255 = log(r2254);
        double r2256 = r2252 * r2255;
        double r2257 = r2251 + r2256;
        double r2258 = t;
        double r2259 = r2257 - r2258;
        return r2259;
}

↓

double f(double x, double y, double z, double t) {
        double r2260 = y;
        double r2261 = cbrt(r2260);
        double r2262 = r2261 * r2261;
        double r2263 = cbrt(r2262);
        double r2264 = log(r2263);
        double r2265 = 2.0;
        double r2266 = r2264 * r2265;
        double r2267 = x;
        double r2268 = r2266 * r2267;
        double r2269 = cbrt(r2261);
        double r2270 = log(r2269);
        double r2271 = r2270 * r2265;
        double r2272 = log(r2261);
        double r2273 = r2271 + r2272;
        double r2274 = r2267 * r2273;
        double r2275 = r2268 + r2274;
        double r2276 = z;
        double r2277 = 1.0;
        double r2278 = log(r2277);
        double r2279 = r2276 * r2278;
        double r2280 = r2276 * r2260;
        double r2281 = r2277 * r2280;
        double r2282 = 0.5;
        double r2283 = pow(r2260, r2265);
        double r2284 = r2276 * r2283;
        double r2285 = pow(r2277, r2265);
        double r2286 = r2284 / r2285;
        double r2287 = r2282 * r2286;
        double r2288 = r2281 + r2287;
        double r2289 = r2279 - r2288;
        double r2290 = r2275 + r2289;
        double r2291 = t;
        double r2292 = r2290 - r2291;
        return r2292;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.6

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

2. Taylor expanded around 0 0.4

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
3. Using strategy rm

4. Applied add-cube-cbrt 0.4

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

5. Applied log-prod 0.4

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

6. Applied distribute-lft-in 0.4

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

7. Simplified 0.4

   \[\leadsto \left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.4

   \[\leadsto \left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

10. Applied cbrt-prod 0.4

    \[\leadsto \left(\left(x \cdot \left(2 \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

11. Applied log-prod 0.4

    \[\leadsto \left(\left(x \cdot \left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

12. Applied distribute-rgt-in 0.4

    \[\leadsto \left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

13. Applied distribute-rgt-in 0.4

    \[\leadsto \left(\left(\color{blue}{\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot x + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right) \cdot x\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

14. Applied associate-+l+ 0.4

    \[\leadsto \left(\color{blue}{\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot x + \left(\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right) \cdot x + x \cdot \log \left(\sqrt[3]{y}\right)\right)\right)} + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

15. Simplified 0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot x + \color{blue}{x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2 + \log \left(\sqrt[3]{y}\right)\right)}\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

16. Final simplification 0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2 + \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))