Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 0.7

Time: 23.7s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r143249 = x;
        double r143250 = y;
        double r143251 = z;
        double r143252 = log(r143251);
        double r143253 = r143250 * r143252;
        double r143254 = t;
        double r143255 = 1.0;
        double r143256 = r143254 - r143255;
        double r143257 = a;
        double r143258 = log(r143257);
        double r143259 = r143256 * r143258;
        double r143260 = r143253 + r143259;
        double r143261 = b;
        double r143262 = r143260 - r143261;
        double r143263 = exp(r143262);
        double r143264 = r143249 * r143263;
        double r143265 = r143264 / r143250;
        return r143265;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r143266 = 1.0;
        double r143267 = sqrt(r143266);
        double r143268 = a;
        double r143269 = cbrt(r143268);
        double r143270 = r143269 * r143269;
        double r143271 = sqrt(r143270);
        double r143272 = r143267 / r143271;
        double r143273 = 1.0;
        double r143274 = pow(r143272, r143273);
        double r143275 = y;
        double r143276 = z;
        double r143277 = r143266 / r143276;
        double r143278 = log(r143277);
        double r143279 = r143275 * r143278;
        double r143280 = r143266 / r143268;
        double r143281 = log(r143280);
        double r143282 = t;
        double r143283 = r143281 * r143282;
        double r143284 = b;
        double r143285 = r143283 + r143284;
        double r143286 = r143279 + r143285;
        double r143287 = exp(r143286);
        double r143288 = sqrt(r143287);
        double r143289 = cbrt(r143288);
        double r143290 = r143289 * r143289;
        double r143291 = r143274 / r143290;
        double r143292 = cbrt(r143275);
        double r143293 = r143292 * r143292;
        double r143294 = r143291 / r143293;
        double r143295 = sqrt(r143269);
        double r143296 = r143267 / r143295;
        double r143297 = pow(r143296, r143273);
        double r143298 = r143297 / r143289;
        double r143299 = r143298 / r143292;
        double r143300 = sqrt(r143268);
        double r143301 = r143266 / r143300;
        double r143302 = pow(r143301, r143273);
        double r143303 = r143302 / r143288;
        double r143304 = x;
        double r143305 = r143303 * r143304;
        double r143306 = r143299 * r143305;
        double r143307 = r143294 * r143306;
        return r143307;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

1. Initial program 2.0

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

2. Taylor expanded around inf 2.0

   \[\leadsto \color{blue}{\frac{x \cdot e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}{y}}\]

3. Simplified 6.2

   \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\frac{y}{x}}}\]
4. Using strategy rm

5. Applied div-inv 6.2

   \[\leadsto \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\color{blue}{y \cdot \frac{1}{x}}}\]

6. Applied add-sqr-sqrt 6.2

   \[\leadsto \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y \cdot \frac{1}{x}}\]

7. Applied add-sqr-sqrt 6.2

   \[\leadsto \frac{\frac{{\left(\frac{1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y \cdot \frac{1}{x}}\]

8. Applied *-un-lft-identity 6.2

   \[\leadsto \frac{\frac{{\left(\frac{\color{blue}{1 \cdot 1}}{\sqrt{a} \cdot \sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y \cdot \frac{1}{x}}\]

9. Applied times-frac 6.2

   \[\leadsto \frac{\frac{{\color{blue}{\left(\frac{1}{\sqrt{a}} \cdot \frac{1}{\sqrt{a}}\right)}}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y \cdot \frac{1}{x}}\]

10. Applied unpow-prod-down 6.2

    \[\leadsto \frac{\frac{\color{blue}{{\left(\frac{1}{\sqrt{a}}\right)}^{1} \cdot {\left(\frac{1}{\sqrt{a}}\right)}^{1}}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y \cdot \frac{1}{x}}\]

11. Applied times-frac 6.2

    \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y \cdot \frac{1}{x}}\]

12. Applied times-frac 1.0

    \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y} \cdot \frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\frac{1}{x}}}\]

13. Simplified 1.0

    \[\leadsto \frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y} \cdot \color{blue}{\left(\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot x\right)}\]
14. Using strategy rm

15. Applied add-cube-cbrt 1.2

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

16. Applied add-cube-cbrt 1.2

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

17. Applied add-cube-cbrt 1.2

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

18. Applied sqrt-prod 1.2

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

19. Applied add-sqr-sqrt 1.2

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

20. Applied times-frac 1.2

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

21. Applied unpow-prod-down 1.2

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

22. Applied times-frac 1.2

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

23. Applied times-frac 1.2

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

24. Applied associate-*l* 0.7

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

25. Final simplification 0.7

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))