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

Average Error: 12.0 → 0.5

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -6.659376642994718 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -0:\\ \;\;\;\;x - {\left(\sqrt[3]{x}\right)}^{2} \cdot \left(\sqrt[3]{x} \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 1.1993304859552876 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (x * ((double) (y - z)))) / y) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (x - ((double) (z * (x / y)))));
	} else {
		double VAR_1;
		if (((((double) (x * ((double) (y - z)))) / y) <= -6.659376642994718e-98)) {
			VAR_1 = (((double) (x * ((double) (y - z)))) / y);
		} else {
			double VAR_2;
			if (((((double) (x * ((double) (y - z)))) / y) <= -0.0)) {
				VAR_2 = ((double) (x - ((double) (((double) pow(((double) cbrt(x)), 2.0)) * ((double) (((double) cbrt(x)) * (z / y)))))));
			} else {
				double VAR_3;
				if (((((double) (x * ((double) (y - z)))) / y) <= 1.1993304859552876e+303)) {
					VAR_3 = (((double) (x * ((double) (y - z)))) / y);
				} else {
					VAR_3 = ((double) (x - ((double) (z * (x / 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	12.0
Target	3.1
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (* x (- y z)) y) < -inf.0 or 1.19933048595528755e303 < (/ (* x (- y z)) y)

   1. Initial program 62.4

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

   2. Simplified 0.4

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

   4. Applied add-cube-cbrt 0.6

      \[\leadsto x - \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \frac{z}{y}\]

   5. Applied associate-*l* 0.6

      \[\leadsto x - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{z}{y}\right)}\]

   6. Simplified 0.6

      \[\leadsto x - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\frac{z}{y} \cdot \sqrt[3]{x}\right)}\]

   7. Taylor expanded around 0 18.7

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

   8. Simplified 0.8

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

   if -inf.0 < (/ (* x (- y z)) y) < -6.65937664299471807e-98 or -0.0 < (/ (* x (- y z)) y) < 1.19933048595528755e303

   1. Initial program 0.4

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

   if -6.65937664299471807e-98 < (/ (* x (- y z)) y) < -0.0

   1. Initial program 17.8

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

   2. Simplified 0.1

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

   4. Applied add-cube-cbrt 0.4

      \[\leadsto x - \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \frac{z}{y}\]

   5. Applied associate-*l* 0.4

      \[\leadsto x - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{z}{y}\right)}\]

   6. Simplified 0.4

      \[\leadsto x - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\frac{z}{y} \cdot \sqrt[3]{x}\right)}\]
   7. Using strategy rm

   8. Applied pow1 0.4

      \[\leadsto x - \left(\sqrt[3]{x} \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{1}}\right) \cdot \left(\frac{z}{y} \cdot \sqrt[3]{x}\right)\]

   9. Applied pow1 0.4

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{1}} \cdot {\left(\sqrt[3]{x}\right)}^{1}\right) \cdot \left(\frac{z}{y} \cdot \sqrt[3]{x}\right)\]

   10. Applied pow-prod-up 0.4

       \[\leadsto x - \color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(1 + 1\right)}} \cdot \left(\frac{z}{y} \cdot \sqrt[3]{x}\right)\]

   11. Simplified 0.4

       \[\leadsto x - {\left(\sqrt[3]{x}\right)}^{\color{blue}{2}} \cdot \left(\frac{z}{y} \cdot \sqrt[3]{x}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -6.659376642994718 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -0:\\ \;\;\;\;x - {\left(\sqrt[3]{x}\right)}^{2} \cdot \left(\sqrt[3]{x} \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 1.1993304859552876 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))