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

Average Error: 12.2 → 0.5

Time: 4.6s

Precision: binary64

Cost: 2765

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -9.510032144855243 \cdot 10^{-194} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{y} \leq 4.963608052259534 \cdot 10^{-269}\right) \land \frac{x \cdot \left(y - z\right)}{y} \leq 9.268358875021113 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (- y z)) y) (- INFINITY))
   (* (- y z) (/ x y))
   (if (or (<= (/ (* x (- y z)) y) -9.510032144855243e-194)
           (and (not (<= (/ (* x (- y z)) y) 4.963608052259534e-269))
                (<= (/ (* x (- y z)) y) 9.268358875021113e+306)))
     (/ (* x (- y z)) y)
     (- x (* z (/ x y))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (((x * (y - z)) / y) <= -((double) INFINITY)) {
		tmp = (y - z) * (x / y);
	} else if ((((x * (y - z)) / y) <= -9.510032144855243e-194) || (!(((x * (y - z)) / y) <= 4.963608052259534e-269) && (((x * (y - z)) / y) <= 9.268358875021113e+306))) {
		tmp = (x * (y - z)) / y;
	} else {
		tmp = x - (z * (x / y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.2
Herbie	0.5

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

Alternatives

Alternative 1
Error	1.8
Cost	59072

\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]

Alternative 2
Error	2.9
Cost	20546

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7656966270300372 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \leq 2.3301905882187746 \cdot 10^{-259}:\\ \;\;\;\;\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{y - z}{\sqrt[3]{y}}\\ \mathbf{elif}\;y \leq 1.4581210858229359 \cdot 10^{-161}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Alternative 3
Error	1.7
Cost	1858

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -9.196833163501908 \cdot 10^{+56}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Alternative 4
Error	0.5
Cost	2577

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty \lor \neg \left(\frac{x \cdot \left(y - z\right)}{y} \leq -8.088983238387148 \cdot 10^{-305} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{y} \leq 0\right) \land \frac{x \cdot \left(y - z\right)}{y} \leq 9.268358875021113 \cdot 10^{+306}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \end{array}\]

Alternative 5
Error	4.5
Cost	3268

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -8.088983238387148 \cdot 10^{-305}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 0:\\ \;\;\;\;\frac{x}{\frac{-y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2.743374945943508 \cdot 10^{+267}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 6
Error	19.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5160633132.16511 \lor \neg \left(z \leq 1.9910782191876272 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 7
Error	20.4
Cost	977

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8411948668968528 \cdot 10^{+106} \lor \neg \left(z \leq 2.3389279572814266 \cdot 10^{-09} \lor \neg \left(z \leq 8.923127825106159 \cdot 10^{+68}\right) \land z \leq 4.0227925167734046 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{x}{\frac{-y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 8
Error	25.7
Cost	64

\[x\]

Alternative 9
Error	61.7
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l*_binary64_21847 0.1

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

   5. Applied associate-/r/_binary64_21848 0.1

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

   6. Simplified 0.1

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

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -9.51003214485524326e-194 or 4.96360805225953432e-269 < (/.f64 (*.f64 x (-.f64 y z)) y) < 9.26835887502111328e306

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   if -9.51003214485524326e-194 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.96360805225953432e-269 or 9.26835887502111328e306 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 38.7

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

   3. Applied associate-/l*_binary64_21847 0.1

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

   4. Taylor expanded around 0 15.9

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

   5. Simplified 1.4

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

   6. Simplified 1.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -9.510032144855243 \cdot 10^{-194} \lor \neg \left(\frac{x \cdot \left(y - z\right)}{y} \leq 4.963608052259534 \cdot 10^{-269}\right) \land \frac{x \cdot \left(y - z\right)}{y} \leq 9.268358875021113 \cdot 10^{+306}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))