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

Average Error: 9.6 → 0.3

Time: 9.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r464383 = x;
        double r464384 = y;
        double r464385 = log(r464384);
        double r464386 = r464383 * r464385;
        double r464387 = z;
        double r464388 = 1.0;
        double r464389 = r464388 - r464384;
        double r464390 = log(r464389);
        double r464391 = r464387 * r464390;
        double r464392 = r464386 + r464391;
        double r464393 = t;
        double r464394 = r464392 - r464393;
        return r464394;
}

↓

double f(double x, double y, double z, double t) {
        double r464395 = y;
        double r464396 = sqrt(r464395);
        double r464397 = sqrt(r464396);
        double r464398 = log(r464397);
        double r464399 = x;
        double r464400 = r464398 * r464399;
        double r464401 = z;
        double r464402 = 1.0;
        double r464403 = log(r464402);
        double r464404 = r464402 * r464395;
        double r464405 = 0.5;
        double r464406 = 2.0;
        double r464407 = pow(r464395, r464406);
        double r464408 = pow(r464402, r464406);
        double r464409 = r464407 / r464408;
        double r464410 = r464405 * r464409;
        double r464411 = r464404 + r464410;
        double r464412 = r464403 - r464411;
        double r464413 = r464401 * r464412;
        double r464414 = log(r464396);
        double r464415 = r464398 + r464414;
        double r464416 = r464399 * r464415;
        double r464417 = r464413 + r464416;
        double r464418 = r464400 + r464417;
        double r464419 = t;
        double r464420 = r464418 - r464419;
        return r464420;
}

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	0.3

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

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 0.3

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

4. Applied add-sqr-sqrt 0.3

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

5. Applied log-prod 0.3

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

6. Applied distribute-lft-in 0.3

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

7. Applied associate-+l+ 0.3

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

8. Simplified 0.3

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

10. Applied add-sqr-sqrt 0.3

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

11. Applied sqrt-prod 0.3

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

12. Applied log-prod 0.3

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

13. Applied distribute-rgt-in 0.3

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

14. Applied associate-+l+ 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020083 
(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))