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

Average Error: 12.0 → 2.6

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.727842335792455 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 1.14248606065761362 \cdot 10^{-294}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{1}{y - z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r744613 = x;
        double r744614 = y;
        double r744615 = z;
        double r744616 = r744614 - r744615;
        double r744617 = r744613 * r744616;
        double r744618 = r744617 / r744614;
        return r744618;
}

↓

double f(double x, double y, double z) {
        double r744619 = x;
        double r744620 = -4.727842335792455e-115;
        bool r744621 = r744619 <= r744620;
        double r744622 = y;
        double r744623 = z;
        double r744624 = r744622 - r744623;
        double r744625 = r744624 / r744622;
        double r744626 = r744619 * r744625;
        double r744627 = 1.1424860606576136e-294;
        bool r744628 = r744619 <= r744627;
        double r744629 = r744619 * r744623;
        double r744630 = r744629 / r744622;
        double r744631 = r744619 - r744630;
        double r744632 = 1.0;
        double r744633 = r744632 / r744624;
        double r744634 = r744622 * r744633;
        double r744635 = r744619 / r744634;
        double r744636 = r744628 ? r744631 : r744635;
        double r744637 = r744621 ? r744626 : r744636;
        return r744637;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.0
Target	3.3
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -4.727842335792455e-115

   1. Initial program 14.6

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

   3. Applied *-un-lft-identity 14.6

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

   4. Applied times-frac 0.9

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

   5. Simplified 0.9

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

   if -4.727842335792455e-115 < x < 1.1424860606576136e-294

   1. Initial program 7.5

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

   3. Applied associate-/l* 7.4

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

   4. Taylor expanded around 0 4.1

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

   if 1.1424860606576136e-294 < x

   1. Initial program 12.1

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

   3. Applied associate-/l* 2.9

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

   5. Applied div-inv 3.0

      \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{1}{y - z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.727842335792455 \cdot 10^{-115}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \le 1.14248606065761362 \cdot 10^{-294}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{1}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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))