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

Average Error: 12.5 → 2.2

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.06095086161022338 \cdot 10^{-199}:\\ \;\;\;\;\frac{x \cdot 1}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \le 6.745538666331437 \cdot 10^{-83}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot \frac{y - z}{y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((y <= -2.0609508616102234e-199)) {
		VAR = ((double) (((double) (x * 1.0)) / ((double) (y / ((double) (y - z))))));
	} else {
		double VAR_1;
		if ((y <= 6.745538666331437e-83)) {
			VAR_1 = ((double) (x - ((double) (((double) (x * z)) / y))));
		} else {
			VAR_1 = ((double) (((double) (x * 1.0)) * ((double) (((double) (y - z)) / y))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.4
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if y < -2.0609508616102234e-199

   1. Initial program 12.2

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

   3. Applied *-un-lft-identity 12.2

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

   4. Applied associate-*r* 12.2

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

   5. Applied associate-/l* 1.6

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

   if -2.0609508616102234e-199 < y < 6.745538666331437e-83

   1. Initial program 10.0

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

   3. Applied *-un-lft-identity 10.0

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

   4. Applied associate-*r* 10.0

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

   5. Applied associate-/l* 10.5

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

   6. Taylor expanded around 0 6.4

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

   if 6.745538666331437e-83 < y

   1. Initial program 14.4

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

   3. Applied *-un-lft-identity 14.4

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

   4. Applied associate-*r* 14.4

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

   5. Applied associate-/l* 0.4

      \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{y}{y - z}}}\]
   6. Using strategy rm

   7. Applied div-inv 0.6

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

   8. Simplified 0.5

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{\frac{y - z}{y}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.06095086161022338 \cdot 10^{-199}:\\ \;\;\;\;\frac{x \cdot 1}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \le 6.745538666331437 \cdot 10^{-83}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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))