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

Average Error: 9.2 → 0.4

Time: 21.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r398576 = x;
        double r398577 = y;
        double r398578 = log(r398577);
        double r398579 = r398576 * r398578;
        double r398580 = z;
        double r398581 = 1.0;
        double r398582 = r398581 - r398577;
        double r398583 = log(r398582);
        double r398584 = r398580 * r398583;
        double r398585 = r398579 + r398584;
        double r398586 = t;
        double r398587 = r398585 - r398586;
        return r398587;
}

↓

double f(double x, double y, double z, double t) {
        double r398588 = 2.0;
        double r398589 = y;
        double r398590 = cbrt(r398589);
        double r398591 = log(r398590);
        double r398592 = r398588 * r398591;
        double r398593 = x;
        double r398594 = r398592 * r398593;
        double r398595 = 1.0;
        double r398596 = r398589 / r398595;
        double r398597 = r398595 / r398589;
        double r398598 = r398596 / r398597;
        double r398599 = -0.5;
        double r398600 = r398598 * r398599;
        double r398601 = log(r398595);
        double r398602 = r398595 * r398589;
        double r398603 = r398601 - r398602;
        double r398604 = r398600 + r398603;
        double r398605 = z;
        double r398606 = r398604 * r398605;
        double r398607 = r398594 + r398606;
        double r398608 = cbrt(r398590);
        double r398609 = log(r398608);
        double r398610 = r398593 * r398609;
        double r398611 = r398588 * r398593;
        double r398612 = r398609 * r398611;
        double r398613 = r398610 + r398612;
        double r398614 = t;
        double r398615 = r398613 - r398614;
        double r398616 = r398607 + r398615;
        return r398616;
}

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. Simplified 9.2

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

3. Taylor expanded around 0 0.3

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

4. Simplified 0.3

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

6. Applied add-cube-cbrt 0.3

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

7. Applied log-prod 0.4

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

8. Applied distribute-lft-in 0.4

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

9. Applied associate--l+ 0.4

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

10. Applied associate-+r+ 0.4

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

11. Simplified 0.4

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

13. Applied add-cube-cbrt 0.4

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

14. Applied log-prod 0.4

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

15. Applied distribute-lft-in 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019194 
(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) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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