Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A

Average Error: 0.6 → 0.5

Time: 4.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r180695 = 1.0;
        double r180696 = x;
        double r180697 = y;
        double r180698 = z;
        double r180699 = r180697 - r180698;
        double r180700 = t;
        double r180701 = r180697 - r180700;
        double r180702 = r180699 * r180701;
        double r180703 = r180696 / r180702;
        double r180704 = r180695 - r180703;
        return r180704;
}

↓

double f(double x, double y, double z, double t) {
        double r180705 = 1.0;
        double r180706 = 1.0;
        double r180707 = cbrt(r180706);
        double r180708 = r180707 * r180707;
        double r180709 = r180708 / r180706;
        double r180710 = x;
        double r180711 = cbrt(r180710);
        double r180712 = r180711 * r180711;
        double r180713 = y;
        double r180714 = t;
        double r180715 = r180713 - r180714;
        double r180716 = cbrt(r180715);
        double r180717 = r180716 * r180716;
        double r180718 = r180712 / r180717;
        double r180719 = z;
        double r180720 = r180713 - r180719;
        double r180721 = r180706 / r180720;
        double r180722 = r180716 / r180711;
        double r180723 = r180721 / r180722;
        double r180724 = r180718 * r180723;
        double r180725 = r180709 * r180724;
        double r180726 = r180705 - r180725;
        return r180726;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 0.6

   \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\]
2. Using strategy rm

3. Applied *-un-lft-identity 0.6

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

4. Applied times-frac 1.1

   \[\leadsto 1 - \color{blue}{\frac{1}{y - z} \cdot \frac{x}{y - t}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 1.1

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

7. Applied add-cube-cbrt 1.1

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

8. Applied times-frac 1.1

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

9. Applied associate-*l* 1.1

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

10. Simplified 1.1

    \[\leadsto 1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{x}{y - t}}{y - z}}\]
11. Using strategy rm

12. Applied clear-num 1.1

    \[\leadsto 1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\color{blue}{\frac{1}{\frac{y - t}{x}}}}{y - z}\]
13. Using strategy rm

14. Applied *-un-lft-identity 1.1

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

15. Applied add-cube-cbrt 1.3

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

16. Applied add-cube-cbrt 1.4

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

17. Applied times-frac 1.4

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

18. Applied add-sqr-sqrt 1.4

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

19. Applied times-frac 1.4

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

20. Applied times-frac 0.5

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

21. Simplified 0.4

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

22. Simplified 0.5

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

23. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1 (/ x (* (- y z) (- y t)))))