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

Average Error: 15.1 → 2.1

Time: 4.0s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y, double z, double t) {
	return ((double) (x * ((double) (((double) (((double) (y / z)) * t)) / t))));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) cbrt(z)))) / ((double) cbrt(((double) cbrt(z)))))) * ((double) (((double) (((double) (((double) cbrt(x)) / ((double) cbrt(z)))) / ((double) cbrt(((double) cbrt(z)))))) * ((double) (y / ((double) cbrt(((double) cbrt(z))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	15.1
Target	1.6
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Initial program 15.1

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

2. Simplified 6.4

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

4. Applied add-cube-cbrt 7.2

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

5. Applied *-un-lft-identity 7.2

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

6. Applied times-frac 7.2

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

7. Applied associate-*r* 5.6

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

8. Simplified 5.6

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

10. Applied add-cube-cbrt 5.7

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

11. Applied cbrt-prod 5.7

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

12. Applied *-un-lft-identity 5.7

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

13. Applied times-frac 5.7

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

14. Applied associate-*r* 6.4

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

15. Simplified 6.4

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

17. Applied cbrt-prod 6.5

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

18. Applied add-cube-cbrt 6.6

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

19. Applied times-frac 6.6

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

20. Applied times-frac 6.6

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

21. Applied associate-*l* 2.1

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

22. Final simplification 2.1

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

Reproduce

herbie shell --seed 2020161 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :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)))