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

Average Error: 9.3 → 0.4

Time: 19.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r24168388 = x;
        double r24168389 = y;
        double r24168390 = log(r24168389);
        double r24168391 = r24168388 * r24168390;
        double r24168392 = z;
        double r24168393 = 1.0;
        double r24168394 = r24168393 - r24168389;
        double r24168395 = log(r24168394);
        double r24168396 = r24168392 * r24168395;
        double r24168397 = r24168391 + r24168396;
        double r24168398 = t;
        double r24168399 = r24168397 - r24168398;
        return r24168399;
}

↓

double f(double x, double y, double z, double t) {
        double r24168400 = y;
        double r24168401 = sqrt(r24168400);
        double r24168402 = cbrt(r24168401);
        double r24168403 = log(r24168402);
        double r24168404 = x;
        double r24168405 = r24168403 * r24168404;
        double r24168406 = 0.6666666666666666;
        double r24168407 = log(r24168400);
        double r24168408 = r24168406 * r24168407;
        double r24168409 = r24168408 + r24168403;
        double r24168410 = r24168404 * r24168409;
        double r24168411 = r24168405 + r24168410;
        double r24168412 = z;
        double r24168413 = 1.0;
        double r24168414 = log(r24168413);
        double r24168415 = r24168400 * r24168413;
        double r24168416 = r24168414 - r24168415;
        double r24168417 = r24168400 / r24168413;
        double r24168418 = 0.5;
        double r24168419 = r24168417 * r24168418;
        double r24168420 = r24168417 * r24168419;
        double r24168421 = r24168416 - r24168420;
        double r24168422 = r24168412 * r24168421;
        double r24168423 = r24168411 + r24168422;
        double r24168424 = t;
        double r24168425 = r24168423 - r24168424;
        return r24168425;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.3
Target	0.2
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.3

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

10. Applied add-sqr-sqrt 0.4

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

11. Applied cbrt-prod 0.4

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

12. Applied log-prod 0.4

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

13. Applied distribute-lft-in 0.4

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

14. Applied associate-+r+ 0.4

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))