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

Average Error: 9.6 → 0.4

Time: 9.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r479784 = x;
        double r479785 = y;
        double r479786 = log(r479785);
        double r479787 = r479784 * r479786;
        double r479788 = z;
        double r479789 = 1.0;
        double r479790 = r479789 - r479785;
        double r479791 = log(r479790);
        double r479792 = r479788 * r479791;
        double r479793 = r479787 + r479792;
        double r479794 = t;
        double r479795 = r479793 - r479794;
        return r479795;
}

↓

double f(double x, double y, double z, double t) {
        double r479796 = x;
        double r479797 = y;
        double r479798 = cbrt(r479797);
        double r479799 = r479798 * r479798;
        double r479800 = log(r479799);
        double r479801 = r479796 * r479800;
        double r479802 = 1.0;
        double r479803 = 0.6666666666666666;
        double r479804 = pow(r479797, r479803);
        double r479805 = r479802 / r479804;
        double r479806 = -0.3333333333333333;
        double r479807 = pow(r479805, r479806);
        double r479808 = r479802 / r479798;
        double r479809 = pow(r479808, r479806);
        double r479810 = r479807 * r479809;
        double r479811 = log(r479810);
        double r479812 = r479811 * r479796;
        double r479813 = z;
        double r479814 = 1.0;
        double r479815 = log(r479814);
        double r479816 = r479813 * r479815;
        double r479817 = r479813 * r479797;
        double r479818 = r479814 * r479817;
        double r479819 = 0.5;
        double r479820 = 2.0;
        double r479821 = pow(r479797, r479820);
        double r479822 = r479813 * r479821;
        double r479823 = pow(r479814, r479820);
        double r479824 = r479822 / r479823;
        double r479825 = r479819 * r479824;
        double r479826 = r479818 + r479825;
        double r479827 = r479816 - r479826;
        double r479828 = r479812 + r479827;
        double r479829 = r479801 + r479828;
        double r479830 = t;
        double r479831 = r479829 - r479830;
        return r479831;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
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 9.6

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

2. Taylor expanded around 0 0.3

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

4. Applied add-cube-cbrt 0.3

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Applied associate-+l+ 0.4

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

8. Simplified 0.4

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

9. Taylor expanded around inf 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Applied add-cube-cbrt 0.4

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

13. Applied times-frac 0.4

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

14. Applied unpow-prod-down 0.4

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

15. Simplified 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020036 
(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))