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

Average Error: 9.0 → 0.4

Time: 32.5s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

↓

\[\left(z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r23496622 = x;
        double r23496623 = y;
        double r23496624 = log(r23496623);
        double r23496625 = r23496622 * r23496624;
        double r23496626 = z;
        double r23496627 = 1.0;
        double r23496628 = r23496627 - r23496623;
        double r23496629 = log(r23496628);
        double r23496630 = r23496626 * r23496629;
        double r23496631 = r23496625 + r23496630;
        double r23496632 = t;
        double r23496633 = r23496631 - r23496632;
        return r23496633;
}

↓

double f(double x, double y, double z, double t) {
        double r23496634 = z;
        double r23496635 = 1.0;
        double r23496636 = log(r23496635);
        double r23496637 = y;
        double r23496638 = r23496635 * r23496637;
        double r23496639 = r23496636 - r23496638;
        double r23496640 = 0.5;
        double r23496641 = r23496635 / r23496637;
        double r23496642 = r23496640 / r23496641;
        double r23496643 = r23496642 / r23496641;
        double r23496644 = r23496639 - r23496643;
        double r23496645 = r23496634 * r23496644;
        double r23496646 = 0.3333333333333333;
        double r23496647 = pow(r23496637, r23496646);
        double r23496648 = log(r23496647);
        double r23496649 = x;
        double r23496650 = r23496648 * r23496649;
        double r23496651 = cbrt(r23496637);
        double r23496652 = log(r23496651);
        double r23496653 = cbrt(r23496651);
        double r23496654 = r23496653 * r23496653;
        double r23496655 = r23496653 * r23496654;
        double r23496656 = log(r23496655);
        double r23496657 = r23496652 + r23496656;
        double r23496658 = r23496649 * r23496657;
        double r23496659 = r23496650 + r23496658;
        double r23496660 = r23496645 + r23496659;
        double r23496661 = t;
        double r23496662 = r23496660 - r23496661;
        return r23496662;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.0
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.0

   \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

2. Taylor expanded around 0 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

3. Simplified 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.3

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

6. Applied log-prod 0.4

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

7. Applied distribute-lft-in 0.4

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

8. Simplified 0.4

   \[\leadsto \left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]
9. Using strategy rm

10. Applied pow1/3 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) + x \cdot \log \color{blue}{\left({y}^{\frac{1}{3}}\right)}\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]
11. Using strategy rm

12. Applied add-cube-cbrt 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right)\right) - t\]

13. Final simplification 0.4

    \[\leadsto \left(z \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019163 
(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) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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