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

Average Error: 9.2 → 0.4

Time: 21.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r305604 = x;
        double r305605 = y;
        double r305606 = log(r305605);
        double r305607 = r305604 * r305606;
        double r305608 = z;
        double r305609 = 1.0;
        double r305610 = r305609 - r305605;
        double r305611 = log(r305610);
        double r305612 = r305608 * r305611;
        double r305613 = r305607 + r305612;
        double r305614 = t;
        double r305615 = r305613 - r305614;
        return r305615;
}

↓

double f(double x, double y, double z, double t) {
        double r305616 = 2.0;
        double r305617 = y;
        double r305618 = 0.6666666666666666;
        double r305619 = pow(r305617, r305618);
        double r305620 = pow(r305619, r305618);
        double r305621 = cbrt(r305617);
        double r305622 = pow(r305621, r305618);
        double r305623 = r305620 * r305622;
        double r305624 = cbrt(r305623);
        double r305625 = cbrt(r305621);
        double r305626 = r305624 * r305625;
        double r305627 = log(r305626);
        double r305628 = r305616 * r305627;
        double r305629 = x;
        double r305630 = r305628 * r305629;
        double r305631 = log(r305621);
        double r305632 = r305629 * r305631;
        double r305633 = r305630 + r305632;
        double r305634 = z;
        double r305635 = 1.0;
        double r305636 = log(r305635);
        double r305637 = r305617 * r305635;
        double r305638 = r305636 - r305637;
        double r305639 = r305634 * r305638;
        double r305640 = 0.5;
        double r305641 = pow(r305617, r305616);
        double r305642 = r305634 * r305641;
        double r305643 = pow(r305635, r305616);
        double r305644 = r305642 / r305643;
        double r305645 = r305640 * r305644;
        double r305646 = r305639 - r305645;
        double r305647 = r305633 + r305646;
        double r305648 = t;
        double r305649 = r305647 - r305648;
        return r305649;
}

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

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.2

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

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

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

10. Applied add-cube-cbrt 0.4

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

11. Applied cbrt-prod 0.4

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

12. Simplified 0.4

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

14. Applied add-cube-cbrt 0.4

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

15. Applied unpow-prod-down 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019305 
(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) (* (/ 0.333333333333333315 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))