Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.1 → 3.3

Time: 29.0s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

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

C

double f(double x, double y, double z, double t) {
        double r30993692 = x;
        double r30993693 = y;
        double r30993694 = z;
        double r30993695 = r30993693 / r30993694;
        double r30993696 = t;
        double r30993697 = r30993695 * r30993696;
        double r30993698 = r30993697 / r30993696;
        double r30993699 = r30993692 * r30993698;
        return r30993699;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r30993700 = y;
        double r30993701 = z;
        double r30993702 = cbrt(r30993701);
        double r30993703 = r30993700 / r30993702;
        double r30993704 = cbrt(r30993703);
        double r30993705 = r30993704 * r30993704;
        double r30993706 = cbrt(r30993702);
        double r30993707 = r30993706 * r30993706;
        double r30993708 = r30993705 / r30993707;
        double r30993709 = cbrt(r30993708);
        double r30993710 = r30993709 * r30993709;
        double r30993711 = r30993702 * r30993702;
        double r30993712 = cbrt(r30993711);
        double r30993713 = cbrt(r30993712);
        double r30993714 = r30993710 / r30993713;
        double r30993715 = r30993709 / r30993713;
        double r30993716 = r30993704 / r30993706;
        double r30993717 = r30993716 / r30993713;
        double r30993718 = x;
        double r30993719 = r30993718 / r30993706;
        double r30993720 = r30993717 * r30993719;
        double r30993721 = r30993715 * r30993720;
        double r30993722 = r30993714 * r30993721;
        return r30993722;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.1
Target	1.5
Herbie	3.3

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Initial program 14.1

   \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

2. Simplified 6.2

   \[\leadsto \color{blue}{y \cdot \frac{x}{z}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 7.0

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

5. Applied *-un-lft-identity 7.0

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

6. Applied times-frac 7.0

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

7. Applied associate-*r* 5.5

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

8. Simplified 5.4

   \[\leadsto \color{blue}{\frac{\frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}}} \cdot \frac{x}{\sqrt[3]{z}}\]
9. Using strategy rm

10. Applied add-cube-cbrt 5.5

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

11. Applied cbrt-prod 5.6

    \[\leadsto \frac{\frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}} \cdot \frac{x}{\color{blue}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}}\]

12. Applied *-un-lft-identity 5.6

    \[\leadsto \frac{\frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}} \cdot \frac{\color{blue}{1 \cdot x}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}\]

13. Applied times-frac 5.6

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

14. Applied associate-*r* 6.2

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

15. Simplified 6.2

    \[\leadsto \color{blue}{\frac{\frac{\frac{y}{\sqrt[3]{z}}}{\sqrt[3]{z}}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}} \cdot \frac{x}{\sqrt[3]{\sqrt[3]{z}}}\]
16. Using strategy rm

17. Applied add-cube-cbrt 6.4

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

18. Applied add-cube-cbrt 6.7

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

19. Applied add-cube-cbrt 6.7

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

20. Applied times-frac 6.7

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

21. Applied times-frac 6.7

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

22. Applied associate-*l* 3.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{z}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{z}}}}{\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{z}}}}{\sqrt[3]{\sqrt[3]{z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}} \cdot \frac{x}{\sqrt[3]{\sqrt[3]{z}}}\right)}\]
23. Using strategy rm

24. Applied add-cube-cbrt 3.3

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

25. Applied times-frac 3.3

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

26. Applied associate-*l* 3.3

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

27. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))