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

Average Error: 12.7 → 2.4

Time: 10.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.580679840002735045330812359603634219556 \cdot 10^{-172}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;x \le 1.524563898325377264226038471226668164466 \cdot 10^{-157}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r508193 = x;
        double r508194 = y;
        double r508195 = z;
        double r508196 = r508194 - r508195;
        double r508197 = r508193 * r508196;
        double r508198 = r508197 / r508194;
        return r508198;
}

↓

double f(double x, double y, double z) {
        double r508199 = x;
        double r508200 = -2.580679840002735e-172;
        bool r508201 = r508199 <= r508200;
        double r508202 = z;
        double r508203 = y;
        double r508204 = r508202 / r508203;
        double r508205 = r508199 * r508204;
        double r508206 = r508199 - r508205;
        double r508207 = 1.5245638983253773e-157;
        bool r508208 = r508199 <= r508207;
        double r508209 = r508199 * r508202;
        double r508210 = r508209 / r508203;
        double r508211 = r508199 - r508210;
        double r508212 = r508203 - r508202;
        double r508213 = r508203 / r508212;
        double r508214 = r508199 / r508213;
        double r508215 = r508208 ? r508211 : r508214;
        double r508216 = r508201 ? r508206 : r508215;
        return r508216;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.7
Target	3.2
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

2. if x < -2.580679840002735e-172

   1. Initial program 13.9

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

   3. Applied associate-/l* 1.2

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

   4. Taylor expanded around 0 4.8

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

   6. Applied *-un-lft-identity 4.8

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

   7. Applied times-frac 1.4

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

   8. Simplified 1.4

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

   if -2.580679840002735e-172 < x < 1.5245638983253773e-157

   1. Initial program 8.9

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

   3. Applied associate-/l* 7.2

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

   4. Taylor expanded around 0 4.6

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

   if 1.5245638983253773e-157 < x

   1. Initial program 14.3

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

   3. Applied associate-/l* 1.7

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.580679840002735045330812359603634219556 \cdot 10^{-172}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;x \le 1.524563898325377264226038471226668164466 \cdot 10^{-157}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e104) (- x (/ (* z x) y)) (if (< z 1.69397660138285259e213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))