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

Average Error: 9.5 → 0.4

Time: 21.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r292502 = x;
        double r292503 = y;
        double r292504 = log(r292503);
        double r292505 = r292502 * r292504;
        double r292506 = z;
        double r292507 = 1.0;
        double r292508 = r292507 - r292503;
        double r292509 = log(r292508);
        double r292510 = r292506 * r292509;
        double r292511 = r292505 + r292510;
        double r292512 = t;
        double r292513 = r292511 - r292512;
        return r292513;
}

↓

double f(double x, double y, double z, double t) {
        double r292514 = 2.0;
        double r292515 = y;
        double r292516 = cbrt(r292515);
        double r292517 = log(r292516);
        double r292518 = r292514 * r292517;
        double r292519 = x;
        double r292520 = r292518 * r292519;
        double r292521 = 0.3333333333333333;
        double r292522 = log(r292515);
        double r292523 = r292522 * r292519;
        double r292524 = r292521 * r292523;
        double r292525 = r292520 + r292524;
        double r292526 = z;
        double r292527 = 1.0;
        double r292528 = log(r292527);
        double r292529 = 0.5;
        double r292530 = r292527 * r292527;
        double r292531 = r292530 / r292515;
        double r292532 = r292529 / r292531;
        double r292533 = r292527 + r292532;
        double r292534 = r292515 * r292533;
        double r292535 = r292528 - r292534;
        double r292536 = r292526 * r292535;
        double r292537 = r292525 + r292536;
        double r292538 = t;
        double r292539 = r292537 - r292538;
        return r292539;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.5
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 9.5

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

2. Taylor expanded around 0 0.3

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

3. Simplified 0.3

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

5. Applied add-cube-cbrt 0.4

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

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

11. Applied pow1/3 0.4

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

12. Applied log-pow 0.4

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

13. Applied associate-*l* 0.4

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

14. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019325 
(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))