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

Average Error: 12.7 → 2.6

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 4.014228494605547884913065641302170952983 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \le 2.367911105194571555458594295483143652103 \cdot 10^{-120}:\\ \;\;\;\;x - \left(x \cdot z\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r589865 = x;
        double r589866 = y;
        double r589867 = z;
        double r589868 = r589866 - r589867;
        double r589869 = r589865 * r589868;
        double r589870 = r589869 / r589866;
        return r589870;
}

↓

double f(double x, double y, double z) {
        double r589871 = y;
        double r589872 = 4.014228494605548e-282;
        bool r589873 = r589871 <= r589872;
        double r589874 = x;
        double r589875 = z;
        double r589876 = r589871 - r589875;
        double r589877 = r589871 / r589876;
        double r589878 = r589874 / r589877;
        double r589879 = 2.3679111051945716e-120;
        bool r589880 = r589871 <= r589879;
        double r589881 = r589874 * r589875;
        double r589882 = 1.0;
        double r589883 = r589882 / r589871;
        double r589884 = r589881 * r589883;
        double r589885 = r589874 - r589884;
        double r589886 = r589876 / r589871;
        double r589887 = r589874 * r589886;
        double r589888 = r589880 ? r589885 : r589887;
        double r589889 = r589873 ? r589878 : r589888;
        return r589889;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.7
Target	2.9
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

2. if y < 4.014228494605548e-282

   1. Initial program 12.6

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

   3. Applied associate-/l* 3.6

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

   if 4.014228494605548e-282 < y < 2.3679111051945716e-120

   1. Initial program 9.1

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

   2. Taylor expanded around 0 5.0

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

   4. Applied div-inv 5.1

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

   if 2.3679111051945716e-120 < y

   1. Initial program 13.8

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

   3. Applied *-un-lft-identity 13.8

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

   4. Applied times-frac 0.8

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

   5. Simplified 0.8

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 4.014228494605547884913065641302170952983 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \le 2.367911105194571555458594295483143652103 \cdot 10^{-120}:\\ \;\;\;\;x - \left(x \cdot z\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))