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

Average Error: 1.5 → 0.6

Time: 4.3s

Precision: 64

Math

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

↓

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

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 ((((cbrt((z - t)) * cbrt((z - t))) / (cbrt((z - a)) * cbrt((z - a)))) * (y / (cbrt((z - a)) / cbrt((z - t))))) + x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.5
Target	1.4
Herbie	0.6

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

Derivation

1. Initial program 1.5

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

2. Simplified 1.5

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

4. Applied clear-num 1.6

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{z - a}{z - t}}}, x\right)\]
5. Using strategy rm

6. Applied fma-udef 1.6

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

7. Simplified 1.4

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

9. Applied add-cube-cbrt 1.9

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

10. Applied add-cube-cbrt 1.8

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

11. Applied times-frac 1.8

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

12. Applied *-un-lft-identity 1.8

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

13. Applied times-frac 0.6

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

14. Simplified 0.6

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

15. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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