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

Average Error: 9.7 → 0.4

Time: 22.9s

Precision: 64

Math

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

↓

\[\left(\left(\left(4 \cdot \left(x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) \cdot 2\right) + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r313421 = x;
        double r313422 = y;
        double r313423 = log(r313422);
        double r313424 = r313421 * r313423;
        double r313425 = z;
        double r313426 = 1.0;
        double r313427 = r313426 - r313422;
        double r313428 = log(r313427);
        double r313429 = r313425 * r313428;
        double r313430 = r313424 + r313429;
        double r313431 = t;
        double r313432 = r313430 - r313431;
        return r313432;
}

↓

double f(double x, double y, double z, double t) {
        double r313433 = 4.0;
        double r313434 = x;
        double r313435 = y;
        double r313436 = cbrt(r313435);
        double r313437 = cbrt(r313436);
        double r313438 = log(r313437);
        double r313439 = r313434 * r313438;
        double r313440 = r313433 * r313439;
        double r313441 = r313438 * r313434;
        double r313442 = 2.0;
        double r313443 = r313441 * r313442;
        double r313444 = r313440 + r313443;
        double r313445 = log(r313436);
        double r313446 = r313445 * r313434;
        double r313447 = r313444 + r313446;
        double r313448 = z;
        double r313449 = 1.0;
        double r313450 = log(r313449);
        double r313451 = r313449 * r313435;
        double r313452 = r313450 - r313451;
        double r313453 = r313448 * r313452;
        double r313454 = 0.5;
        double r313455 = pow(r313435, r313442);
        double r313456 = r313448 * r313455;
        double r313457 = pow(r313449, r313442);
        double r313458 = r313456 / r313457;
        double r313459 = r313454 * r313458;
        double r313460 = r313453 - r313459;
        double r313461 = r313447 + r313460;
        double r313462 = t;
        double r313463 = r313461 - r313462;
        return r313463;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.7
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 9.7

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

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. Simplified 0.3

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

5. Applied add-cube-cbrt 0.3

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Applied log-prod 0.4

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

13. Applied distribute-lft-in 0.4

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

14. Applied distribute-lft-in 0.4

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

15. Simplified 0.4

    \[\leadsto \left(\left(\left(\color{blue}{4 \cdot \left(x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + x \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

16. Simplified 0.4

    \[\leadsto \left(\left(\left(4 \cdot \left(x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) \cdot 2}\right) + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

17. Final simplification 0.4

    \[\leadsto \left(\left(\left(4 \cdot \left(x \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot x\right) \cdot 2\right) + \log \left(\sqrt[3]{y}\right) \cdot x\right) + \left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\]

Reproduce

herbie shell --seed 2019323 
(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))