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

Average Error: 9.6 → 0.4

Time: 23.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r20423428 = x;
        double r20423429 = y;
        double r20423430 = log(r20423429);
        double r20423431 = r20423428 * r20423430;
        double r20423432 = z;
        double r20423433 = 1.0;
        double r20423434 = r20423433 - r20423429;
        double r20423435 = log(r20423434);
        double r20423436 = r20423432 * r20423435;
        double r20423437 = r20423431 + r20423436;
        double r20423438 = t;
        double r20423439 = r20423437 - r20423438;
        return r20423439;
}

↓

double f(double x, double y, double z, double t) {
        double r20423440 = z;
        double r20423441 = 1.0;
        double r20423442 = log(r20423441);
        double r20423443 = y;
        double r20423444 = r20423443 * r20423441;
        double r20423445 = r20423442 - r20423444;
        double r20423446 = r20423443 / r20423441;
        double r20423447 = r20423446 * r20423446;
        double r20423448 = 0.5;
        double r20423449 = r20423447 * r20423448;
        double r20423450 = r20423445 - r20423449;
        double r20423451 = r20423440 * r20423450;
        double r20423452 = cbrt(r20423443);
        double r20423453 = log(r20423452);
        double r20423454 = x;
        double r20423455 = r20423453 * r20423454;
        double r20423456 = r20423451 + r20423455;
        double r20423457 = 0.6666666666666666;
        double r20423458 = pow(r20423443, r20423457);
        double r20423459 = log(r20423458);
        double r20423460 = r20423459 * r20423454;
        double r20423461 = r20423456 + r20423460;
        double r20423462 = t;
        double r20423463 = r20423461 - r20423462;
        return r20423463;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
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 9.6

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

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. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-rgt-in 0.4

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

8. Applied associate-+l+ 0.4

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

10. Applied pow1/3 0.4

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

11. Applied pow1/3 0.4

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

12. Applied pow-prod-up 0.4

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

13. Simplified 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019174 
(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))