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

Average Error: 10.7 → 0.5

Time: 4.1s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.7
Target	1.4
Herbie	0.5

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

Derivation

1. Initial program 10.7

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

2. Simplified 3.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)}\]
3. Using strategy rm

4. Applied fma-udef 3.1

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

6. Applied div-inv 3.1

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

7. Applied associate-*l* 1.6

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

8. Simplified 1.5

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

10. Applied add-cube-cbrt 2.0

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

11. Applied add-cube-cbrt 1.9

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

12. Applied times-frac 1.9

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

13. Applied associate-*r* 0.5

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

14. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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