Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.0 → 1.0

Time: 3.6s

Precision: binary64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.0
Target	2.1
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < -0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Initial program 2.0

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

3. Applied add-cube-cbrt 2.5

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

4. Applied *-un-lft-identity 2.5

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

5. Applied times-frac 2.5

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

6. Applied associate-*r* 4.6

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

7. Simplified 4.6

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

9. Applied *-un-lft-identity 4.6

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

10. Applied cbrt-prod 4.6

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

11. Applied add-cube-cbrt 4.8

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

12. Applied times-frac 4.8

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

13. Applied associate-*r* 4.5

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

14. Simplified 0.9

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

16. Applied add-cube-cbrt 1.0

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

17. Simplified 1.0

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

18. Simplified 1.0

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

19. Final simplification 1.0

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

Reproduce

herbie shell --seed 2020199 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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