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

Average Error: 10.8 → 2.1

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.58682015110359294 \cdot 10^{-296}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((x <= 1.586820151103593e-296)) {
		VAR = (x + (((y - z) / (a - z)) * t));
	} else {
		VAR = (x + ((y - z) / ((a - z) / t)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.8
Target	0.6
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if x < 1.586820151103593e-296

   1. Initial program 10.6

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

   3. Applied associate-/l* 3.1

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

   5. Applied associate-/r/ 1.2

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

   if 1.586820151103593e-296 < x

   1. Initial program 10.9

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

   3. Applied associate-/l* 3.0

      \[\leadsto x + \color{blue}{\frac{y - z}{\frac{a - z}{t}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.58682015110359294 \cdot 10^{-296}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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))))