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

Average Error: 2.0 → 1.9

Time: 5.2s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.0
Target	2.2
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

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

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

4. Applied associate-*l* 2.5

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

6. Applied associate-*l* 2.5

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

8. Applied associate-*r/ 2.9

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

10. Applied pow1 2.9

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

11. Applied pow1 2.9

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

12. Applied pow-prod-down 2.9

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

13. Applied pow1 2.9

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

14. Applied pow-prod-down 2.9

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

15. Simplified 1.9

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

16. Final simplification 1.9

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

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)) -1013646692435.887) (+ 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))))