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

Average Error: 10.2 → 0.8

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -4.778727330201435 \cdot 10^{208}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 2.0736905137245513 \cdot 10^{255}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t} - \frac{z}{t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z)))) <= -4.778727330201435e+208)) {
		VAR = ((double) (x + ((double) (((double) (y - z)) * ((double) (t / ((double) (a - z))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z)))) <= 2.0736905137245513e+255)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z))))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (y - z)) / ((double) (((double) (a / t)) - ((double) (z / t))))))));
		}
		VAR = VAR_1;
	}
	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	10.2
Target	0.8
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/ (* (- y z) t) (- a z)) < -4.778727330201435e208

   1. Initial program 48.4

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

   3. Applied *-un-lft-identity 48.4

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

   4. Applied times-frac 3.0

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

   5. Simplified 3.0

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

   if -4.778727330201435e208 < (/ (* (- y z) t) (- a z)) < 2.0736905137245513e255

   1. Initial program 0.3

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

   if 2.0736905137245513e255 < (/ (* (- y z) t) (- a z))

   1. Initial program 54.4

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

   3. Applied associate-/l* 2.5

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

   5. Applied div-sub 2.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \le -4.778727330201435 \cdot 10^{208}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \le 2.0736905137245513 \cdot 10^{255}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t} - \frac{z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))))