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

Average Error: 11.9 → 1.5

Time: 10.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.998165745453842247107774064380461371887 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;y \le 9.436120878493814052713516163817953193206 \cdot 10^{-81}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r39239087 = x;
        double r39239088 = y;
        double r39239089 = z;
        double r39239090 = r39239088 - r39239089;
        double r39239091 = r39239087 * r39239090;
        double r39239092 = r39239091 / r39239088;
        return r39239092;
}

↓

double f(double x, double y, double z) {
        double r39239093 = y;
        double r39239094 = -1.9981657454538422e-24;
        bool r39239095 = r39239093 <= r39239094;
        double r39239096 = x;
        double r39239097 = z;
        double r39239098 = r39239093 - r39239097;
        double r39239099 = r39239098 / r39239093;
        double r39239100 = r39239096 * r39239099;
        double r39239101 = 9.436120878493814e-81;
        bool r39239102 = r39239093 <= r39239101;
        double r39239103 = r39239096 * r39239097;
        double r39239104 = r39239103 / r39239093;
        double r39239105 = r39239096 - r39239104;
        double r39239106 = r39239093 / r39239098;
        double r39239107 = r39239096 / r39239106;
        double r39239108 = r39239102 ? r39239105 : r39239107;
        double r39239109 = r39239095 ? r39239100 : r39239108;
        return r39239109;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	11.9
Target	3.1
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1.9981657454538422e-24

   1. Initial program 14.8

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

   3. Applied *-un-lft-identity 14.8

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if -1.9981657454538422e-24 < y < 9.436120878493814e-81

   1. Initial program 7.3

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

   3. Applied associate-/l* 7.9

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

   4. Taylor expanded around 0 4.2

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

   if 9.436120878493814e-81 < y

   1. Initial program 13.6

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

   3. Applied associate-/l* 0.3

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.998165745453842247107774064380461371887 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;y \le 9.436120878493814052713516163817953193206 \cdot 10^{-81}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"

  :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))