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

Average Error: 11.1 → 1.4

Time: 10.1s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

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

↓

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) ((((double) (y - z)) / ((double) (a - 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	11.1
Target	0.6
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Initial program 11.1

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

3. Applied *-un-lft-identity 11.1

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

4. Applied times-frac 3.2

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

5. Simplified 3.2

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

7. Applied pow1 3.2

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

8. Applied pow1 3.2

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

9. Applied pow-prod-down 3.2

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

10. Simplified 1.4

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

11. Final simplification 1.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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