Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 25.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r85392 = x;
        double r85393 = y;
        double r85394 = log(r85393);
        double r85395 = r85392 * r85394;
        double r85396 = r85395 - r85393;
        double r85397 = z;
        double r85398 = r85396 - r85397;
        double r85399 = t;
        double r85400 = log(r85399);
        double r85401 = r85398 + r85400;
        return r85401;
}

↓

double f(double x, double y, double z, double t) {
        double r85402 = x;
        double r85403 = y;
        double r85404 = 0.6666666666666666;
        double r85405 = pow(r85403, r85404);
        double r85406 = cbrt(r85405);
        double r85407 = r85406 * r85406;
        double r85408 = cbrt(r85403);
        double r85409 = cbrt(r85408);
        double r85410 = 2.0;
        double r85411 = pow(r85409, r85410);
        double r85412 = r85407 * r85411;
        double r85413 = log(r85412);
        double r85414 = r85402 * r85413;
        double r85415 = log(r85408);
        double r85416 = r85415 * r85402;
        double r85417 = r85416 - r85403;
        double r85418 = z;
        double r85419 = r85417 - r85418;
        double r85420 = r85414 + r85419;
        double r85421 = t;
        double r85422 = log(r85421);
        double r85423 = r85420 + r85422;
        return r85423;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 0.1

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate--l+ 0.1

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

7. Applied associate--l+ 0.1

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

8. Simplified 0.1

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

10. Applied add-cube-cbrt 0.1

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

11. Applied cbrt-prod 0.1

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

12. Applied add-cube-cbrt 0.1

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

13. Applied cbrt-prod 0.1

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

14. Applied swap-sqr 0.1

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

15. Simplified 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))