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

Average Error: 9.8 → 0.4

Time: 14.5s

Precision: 64

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

Math

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

↓

\[\left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r504411 = x;
        double r504412 = y;
        double r504413 = log(r504412);
        double r504414 = r504411 * r504413;
        double r504415 = z;
        double r504416 = 1.0;
        double r504417 = r504416 - r504412;
        double r504418 = log(r504417);
        double r504419 = r504415 * r504418;
        double r504420 = r504414 + r504419;
        double r504421 = t;
        double r504422 = r504420 - r504421;
        return r504422;
}

↓

double f(double x, double y, double z, double t) {
        double r504423 = x;
        double r504424 = y;
        double r504425 = cbrt(r504424);
        double r504426 = 1.6666666666666667;
        double r504427 = pow(r504425, r504426);
        double r504428 = cbrt(r504425);
        double r504429 = r504427 * r504428;
        double r504430 = log(r504429);
        double r504431 = r504423 * r504430;
        double r504432 = 0.3333333333333333;
        double r504433 = pow(r504424, r504432);
        double r504434 = log(r504433);
        double r504435 = r504423 * r504434;
        double r504436 = z;
        double r504437 = 1.0;
        double r504438 = log(r504437);
        double r504439 = r504437 * r504424;
        double r504440 = 0.5;
        double r504441 = 2.0;
        double r504442 = pow(r504424, r504441);
        double r504443 = pow(r504437, r504441);
        double r504444 = r504442 / r504443;
        double r504445 = r504440 * r504444;
        double r504446 = r504439 + r504445;
        double r504447 = r504438 - r504446;
        double r504448 = r504436 * r504447;
        double r504449 = r504435 + r504448;
        double r504450 = r504431 + r504449;
        double r504451 = t;
        double r504452 = r504450 - r504451;
        return r504452;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.8
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 9.8

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

2. Taylor expanded around 0 0.4

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

4. Applied add-cube-cbrt 0.4

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Applied associate-+l+ 0.4

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

9. Applied pow1/3 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Applied associate-*r* 0.4

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

13. Simplified 0.4

    \[\leadsto \left(x \cdot \log \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{\frac{5}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

14. Final simplification 0.4

    \[\leadsto \left(x \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(x \cdot \log \left({y}^{\frac{1}{3}}\right) + z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020047 
(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))