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

Average Error: 9.6 → 0.4

Time: 3.5m

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

↓

\[\left(z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right) + \left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{y}\right)\right)\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r31422367 = x;
        double r31422368 = y;
        double r31422369 = log(r31422368);
        double r31422370 = r31422367 * r31422369;
        double r31422371 = z;
        double r31422372 = 1.0;
        double r31422373 = r31422372 - r31422368;
        double r31422374 = log(r31422373);
        double r31422375 = r31422371 * r31422374;
        double r31422376 = r31422370 + r31422375;
        double r31422377 = t;
        double r31422378 = r31422376 - r31422377;
        return r31422378;
}

↓

double f(double x, double y, double z, double t) {
        double r31422379 = z;
        double r31422380 = 1.0;
        double r31422381 = log(r31422380);
        double r31422382 = 0.5;
        double r31422383 = y;
        double r31422384 = r31422380 / r31422383;
        double r31422385 = r31422382 / r31422384;
        double r31422386 = r31422385 / r31422384;
        double r31422387 = r31422380 * r31422383;
        double r31422388 = r31422386 + r31422387;
        double r31422389 = r31422381 - r31422388;
        double r31422390 = r31422379 * r31422389;
        double r31422391 = x;
        double r31422392 = cbrt(r31422383);
        double r31422393 = log(r31422392);
        double r31422394 = r31422391 * r31422393;
        double r31422395 = cbrt(r31422392);
        double r31422396 = log(r31422395);
        double r31422397 = r31422396 * r31422391;
        double r31422398 = r31422394 + r31422397;
        double r31422399 = r31422395 * r31422395;
        double r31422400 = log(r31422399);
        double r31422401 = r31422400 + r31422393;
        double r31422402 = r31422391 * r31422401;
        double r31422403 = r31422398 + r31422402;
        double r31422404 = r31422390 + r31422403;
        double r31422405 = t;
        double r31422406 = r31422404 - r31422405;
        return r31422406;
}

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

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{\frac{1}{3}}{1.0 \cdot \left(1.0 \cdot 1.0\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.6

   \[\left(x \cdot \log y + z \cdot \log \left(1.0 - y\right)\right) - t\]

2. Taylor expanded around 0 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

3. Simplified 0.3

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.4

   \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

6. Applied log-prod 0.4

   \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

7. Applied distribute-lft-in 0.4

   \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

8. Simplified 0.4

   \[\leadsto \left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]
9. Using strategy rm

10. Applied add-cube-cbrt 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

11. Applied log-prod 0.4

    \[\leadsto \left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

12. Applied associate-+r+ 0.4

    \[\leadsto \left(\left(x \cdot \color{blue}{\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

13. Applied distribute-rgt-in 0.4

    \[\leadsto \left(\left(\color{blue}{\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

14. Applied associate-+l+ 0.4

    \[\leadsto \left(\color{blue}{\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot x + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x + x \cdot \log \left(\sqrt[3]{y}\right)\right)\right)} + z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right)\right) - t\]

15. Final simplification 0.4

    \[\leadsto \left(z \cdot \left(\log 1.0 - \left(\frac{\frac{\frac{1}{2}}{\frac{1.0}{y}}}{\frac{1.0}{y}} + 1.0 \cdot y\right)\right) + \left(\left(x \cdot \log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) + x \cdot \left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{y}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019158 
(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) (* (/ 1/3 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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