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

Average Error: 14.6 → 2.7

Time: 2.6m

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r7342982 = x;
        double r7342983 = y;
        double r7342984 = z;
        double r7342985 = r7342983 / r7342984;
        double r7342986 = t;
        double r7342987 = r7342985 * r7342986;
        double r7342988 = r7342987 / r7342986;
        double r7342989 = r7342982 * r7342988;
        return r7342989;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r7342990 = y;
        double r7342991 = z;
        double r7342992 = cbrt(r7342991);
        double r7342993 = r7342990 / r7342992;
        double r7342994 = r7342992 * r7342992;
        double r7342995 = cbrt(r7342994);
        double r7342996 = x;
        double r7342997 = cbrt(r7342996);
        double r7342998 = r7342997 * r7342997;
        double r7342999 = r7342995 / r7342998;
        double r7343000 = r7342993 / r7342999;
        double r7343001 = r7342997 / r7342992;
        double r7343002 = 1.0;
        double r7343003 = cbrt(r7342992);
        double r7343004 = r7343002 / r7343003;
        double r7343005 = r7343001 * r7343004;
        double r7343006 = r7343000 * r7343005;
        return r7343006;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 14.6

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

2. Simplified 6.2

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

4. Applied add-cube-cbrt 7.0

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

5. Applied *-un-lft-identity 7.0

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

6. Applied times-frac 7.0

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

7. Applied associate-*l* 5.3

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

9. Applied associate-*l/ 6.5

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

10. Applied associate-*r/ 6.5

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

11. Simplified 5.3

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

13. Applied *-un-lft-identity 5.3

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

14. Applied times-frac 4.8

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

15. Applied associate-/l* 5.7

    \[\leadsto \color{blue}{\frac{\frac{\frac{y}{\sqrt[3]{z}}}{1}}{\frac{\sqrt[3]{z}}{\frac{x}{\sqrt[3]{z}}}}}\]
16. Using strategy rm

17. Applied *-un-lft-identity 5.7

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

18. Applied add-cube-cbrt 5.9

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

19. Applied times-frac 5.9

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

20. Applied add-cube-cbrt 5.9

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

21. Applied cbrt-prod 6.0

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

22. Applied times-frac 6.0

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

23. Applied div-inv 6.0

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

24. Applied times-frac 2.7

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

25. Simplified 2.7

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

26. Simplified 2.7

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

27. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))