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

Average Error: 9.2 → 0.4

Time: 12.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x \cdot 4\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)\right) + \frac{-1}{3} \cdot \left(\left(-\log y\right) \cdot x\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r539514 = x;
        double r539515 = y;
        double r539516 = log(r539515);
        double r539517 = r539514 * r539516;
        double r539518 = z;
        double r539519 = 1.0;
        double r539520 = r539519 - r539515;
        double r539521 = log(r539520);
        double r539522 = r539518 * r539521;
        double r539523 = r539517 + r539522;
        double r539524 = t;
        double r539525 = r539523 - r539524;
        return r539525;
}

↓

double f(double x, double y, double z, double t) {
        double r539526 = y;
        double r539527 = cbrt(r539526);
        double r539528 = cbrt(r539527);
        double r539529 = log(r539528);
        double r539530 = x;
        double r539531 = 4.0;
        double r539532 = r539530 * r539531;
        double r539533 = r539529 * r539532;
        double r539534 = 2.0;
        double r539535 = r539529 * r539534;
        double r539536 = r539530 * r539535;
        double r539537 = r539533 + r539536;
        double r539538 = -0.3333333333333333;
        double r539539 = log(r539526);
        double r539540 = -r539539;
        double r539541 = r539540 * r539530;
        double r539542 = r539538 * r539541;
        double r539543 = r539537 + r539542;
        double r539544 = z;
        double r539545 = 1.0;
        double r539546 = log(r539545);
        double r539547 = r539545 * r539526;
        double r539548 = r539546 - r539547;
        double r539549 = r539544 * r539548;
        double r539550 = 0.5;
        double r539551 = pow(r539526, r539534);
        double r539552 = r539544 * r539551;
        double r539553 = pow(r539545, r539534);
        double r539554 = r539552 / r539553;
        double r539555 = r539550 * r539554;
        double r539556 = r539549 - r539555;
        double r539557 = r539543 + r539556;
        double r539558 = t;
        double r539559 = r539557 - r539558;
        return r539559;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.2
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 9.2

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

9. Taylor expanded around inf 0.4

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

10. Simplified 0.3

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

12. Applied add-cube-cbrt 0.3

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

13. Applied log-prod 0.4

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

14. Applied distribute-lft-in 0.4

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

15. Applied distribute-lft-in 0.4

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

16. Simplified 0.4

    \[\leadsto \left(\left(\left(\color{blue}{\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x \cdot 4\right)} + x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \frac{-1}{3} \cdot \left(\left(-\log y\right) \cdot x\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

17. Simplified 0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x \cdot 4\right) + \color{blue}{x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)}\right) + \frac{-1}{3} \cdot \left(\left(-\log y\right) \cdot x\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

18. Final simplification 0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x \cdot 4\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right)\right) + \frac{-1}{3} \cdot \left(\left(-\log y\right) \cdot x\right)\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

Reproduce

herbie shell --seed 2020045 
(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))