Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Error: 11.8 → 0.5

Time: 2.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \le -3.5302740687576495 \cdot 10^{307}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le -1.6831734572194845 \cdot 10^{59}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(y - z\right)}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 9.0428667222306372 \cdot 10^{60}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 7.4856792036076336 \cdot 10^{286}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((x * (y - z)) / y) <= -3.5302740687576495e+307)) {
		VAR = (x * ((y - z) / y));
	} else {
		double VAR_1;
		if ((((x * (y - z)) / y) <= -1.6831734572194845e+59)) {
			VAR_1 = (1.0 / (y / (x * (y - z))));
		} else {
			double VAR_2;
			if ((((x * (y - z)) / y) <= 9.042866722230637e+60)) {
				VAR_2 = (x / (y / (y - z)));
			} else {
				double VAR_3;
				if ((((x * (y - z)) / y) <= 7.485679203607634e+286)) {
					VAR_3 = (1.0 / (y / (x * (y - z))));
				} else {
					VAR_3 = (x * ((y - z) / y));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	11.8
Target	3.0
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (* x (- y z)) y) < -3.5302740687576495e+307 or 7.485679203607634e+286 < (/ (* x (- y z)) y)

   1. Initial program 59.6

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

   3. Applied *-un-lft-identity 59.6

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

   4. Applied times-frac 1.1

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

   5. Simplified 1.1

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

   if -3.5302740687576495e+307 < (/ (* x (- y z)) y) < -1.6831734572194845e+59 or 9.042866722230637e+60 < (/ (* x (- y z)) y) < 7.485679203607634e+286

   1. Initial program 0.2

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

   3. Applied clear-num 0.3

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

   if -1.6831734572194845e+59 < (/ (* x (- y z)) y) < 9.042866722230637e+60

   1. Initial program 5.5

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

   3. Applied associate-/l* 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \le -3.5302740687576495 \cdot 10^{307}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le -1.6831734572194845 \cdot 10^{59}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(y - z\right)}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 9.0428667222306372 \cdot 10^{60}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 7.4856792036076336 \cdot 10^{286}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(y - z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))