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

Average Error: 12.2 → 1.6

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -305510528796828566000 \lor \neg \left(y \le 4.2929037794214209 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \end{array}\]

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 <= -3.0551052879682857e+20) || !(y <= 4.292903779421421e-90))) {
		VAR = ((double) (x / ((double) (y / ((double) (y - z))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (x * z)) / y))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.2
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if y < -3.0551052879682857e+20 or 4.292903779421421e-90 < y

   1. Initial program 15.3

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

   3. Applied associate-/l* 0.2

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

   if -3.0551052879682857e+20 < y < 4.292903779421421e-90

   1. Initial program 7.0

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

   3. Applied associate-/l* 7.9

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

   4. Taylor expanded around 0 4.0

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -305510528796828566000 \lor \neg \left(y \le 4.2929037794214209 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))