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

Average Error: 12.5 → 0.9

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -4.024304033415889 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 6.213263482840311 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{\frac{1}{1 - \frac{z}{y}}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 4.6064334916349155 \cdot 10^{+219}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (- y z)) y) (- INFINITY))
   (* x (/ (- y z) y))
   (if (<= (/ (* x (- y z)) y) -4.024304033415889e+43)
     (/ (* x (- y z)) y)
     (if (<= (/ (* x (- y z)) y) 6.213263482840311e-79)
       (/ x (/ 1.0 (- 1.0 (/ z y))))
       (if (<= (/ (* x (- y z)) y) 4.6064334916349155e+219)
         (- x (/ (* x z) y))
         (* x (/ (- y z) 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 = x * ((y - z) / y);
	} else if (((x * (y - z)) / y) <= -4.024304033415889e+43) {
		tmp = (x * (y - z)) / y;
	} else if (((x * (y - z)) / y) <= 6.213263482840311e-79) {
		tmp = x / (1.0 / (1.0 - (z / y)));
	} else if (((x * (y - z)) / y) <= 4.6064334916349155e+219) {
		tmp = x - ((x * z) / y);
	} else {
		tmp = x * ((y - z) / y);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.1
Herbie	0.9

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -inf.0 or 4.60643349163491549e219 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 49.3

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

   3. Applied *-un-lft-identity_binary64_20197 49.3

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

   4. Applied times-frac_binary64_20203 3.3

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

   5. Simplified 3.3

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

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -4.02430403341588877e43

   1. Initial program 0.2

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

   if -4.02430403341588877e43 < (/.f64 (*.f64 x (-.f64 y z)) y) < 6.21326348284031129e-79

   1. Initial program 6.7

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

   3. Applied associate-/l*_binary64_20142 0.2

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

   5. Applied clear-num_binary64_20196 0.3

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

   6. Simplified 0.3

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

   if 6.21326348284031129e-79 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.60643349163491549e219

   1. Initial program 0.4

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

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{y}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -4.024304033415889 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 6.213263482840311 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{\frac{1}{1 - \frac{z}{y}}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 4.6064334916349155 \cdot 10^{+219}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))