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

Average Error: 9.0 → 0.5

Time: 7.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r369463 = x;
        double r369464 = y;
        double r369465 = log(r369464);
        double r369466 = r369463 * r369465;
        double r369467 = z;
        double r369468 = 1.0;
        double r369469 = r369468 - r369464;
        double r369470 = log(r369469);
        double r369471 = r369467 * r369470;
        double r369472 = r369466 + r369471;
        double r369473 = t;
        double r369474 = r369472 - r369473;
        return r369474;
}

↓

double f(double x, double y, double z, double t) {
        double r369475 = z;
        double r369476 = 1.0;
        double r369477 = log(r369476);
        double r369478 = r369475 * r369477;
        double r369479 = 0.6666666666666666;
        double r369480 = y;
        double r369481 = log(r369480);
        double r369482 = x;
        double r369483 = r369481 * r369482;
        double r369484 = r369479 * r369483;
        double r369485 = cbrt(r369480);
        double r369486 = log(r369485);
        double r369487 = r369486 * r369482;
        double r369488 = r369484 + r369487;
        double r369489 = r369478 + r369488;
        double r369490 = r369475 * r369480;
        double r369491 = r369476 * r369490;
        double r369492 = r369489 - r369491;
        double r369493 = t;
        double r369494 = r369492 - r369493;
        return r369494;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.0
Target	0.2
Herbie	0.5

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.0

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

2. Taylor expanded around 0 0.5

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

4. Applied add-cube-cbrt 0.5

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

5. Applied log-prod 0.5

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

6. Applied distribute-lft-in 0.5

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

7. Simplified 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied cbrt-prod 0.5

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

11. Applied log-prod 0.5

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

12. Applied distribute-rgt-in 0.5

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

13. Applied associate-+r+ 0.5

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

14. Simplified 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020062 
(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))