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

Average Error: 12.4 → 1.8

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.290655561206783908563838071823474030955 \cdot 10^{97}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{elif}\;x \le 2.731727571395544206998641716133386673057 \cdot 10^{-15}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{-y}{-\left(y - z\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r497698 = x;
        double r497699 = y;
        double r497700 = z;
        double r497701 = r497699 - r497700;
        double r497702 = r497698 * r497701;
        double r497703 = r497702 / r497699;
        return r497703;
}

↓

double f(double x, double y, double z) {
        double r497704 = x;
        double r497705 = -1.290655561206784e+97;
        bool r497706 = r497704 <= r497705;
        double r497707 = 1.0;
        double r497708 = z;
        double r497709 = y;
        double r497710 = r497708 / r497709;
        double r497711 = r497707 - r497710;
        double r497712 = r497704 * r497711;
        double r497713 = 2.7317275713955442e-15;
        bool r497714 = r497704 <= r497713;
        double r497715 = r497704 * r497708;
        double r497716 = r497715 / r497709;
        double r497717 = r497704 - r497716;
        double r497718 = -r497709;
        double r497719 = r497709 - r497708;
        double r497720 = -r497719;
        double r497721 = r497718 / r497720;
        double r497722 = r497704 / r497721;
        double r497723 = r497714 ? r497717 : r497722;
        double r497724 = r497706 ? r497712 : r497723;
        return r497724;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.3
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if x < -1.290655561206784e+97

   1. Initial program 31.2

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

   3. Applied associate-/l* 0.1

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

   5. Applied div-inv 0.1

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

   6. Simplified 0.1

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

   if -1.290655561206784e+97 < x < 2.7317275713955442e-15

   1. Initial program 5.6

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

   2. Taylor expanded around 0 2.7

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

   if 2.7317275713955442e-15 < x

   1. Initial program 20.8

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

   3. Applied associate-/l* 0.1

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

   5. Applied frac-2neg 0.1

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.290655561206783908563838071823474030955 \cdot 10^{97}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{elif}\;x \le 2.731727571395544206998641716133386673057 \cdot 10^{-15}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{-y}{-\left(y - z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))