Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B

Average Error: 11.0 → 0.8

Time: 4.8s

Precision: binary64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	11.0
Target	1.2
Herbie	0.8

\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

1. Initial program Error: 11.0 bits

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

2. Simplified Error: 1.3 bits

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

4. Applied add-cube-cbrt Error: 1.8 bits

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

5. Applied add-cube-cbrt Error: 1.6 bits

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

6. Applied times-frac Error: 1.6 bits

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

7. Applied associate-*r* Error: 0.5 bits

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

8. Simplified Error: 0.6 bits

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

10. Applied add-cube-cbrt Error: 0.8 bits

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

11. Applied associate-*l* Error: 0.8 bits

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

12. Simplified Error: 0.8 bits

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

13. Final simplification Error: 0.8 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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