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

Average Error: 12.5 → 2.7

Time: 14.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.164590204538870698162352482223739848451 \cdot 10^{-122} \lor \neg \left(x \le 1.503950824539007656103296067322360995492 \cdot 10^{-300}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r433868 = x;
        double r433869 = y;
        double r433870 = z;
        double r433871 = r433869 - r433870;
        double r433872 = r433868 * r433871;
        double r433873 = r433872 / r433869;
        return r433873;
}

↓

double f(double x, double y, double z) {
        double r433874 = x;
        double r433875 = -1.1645902045388707e-122;
        bool r433876 = r433874 <= r433875;
        double r433877 = 1.5039508245390077e-300;
        bool r433878 = r433874 <= r433877;
        double r433879 = !r433878;
        bool r433880 = r433876 || r433879;
        double r433881 = 1.0;
        double r433882 = z;
        double r433883 = y;
        double r433884 = r433882 / r433883;
        double r433885 = r433881 - r433884;
        double r433886 = r433874 * r433885;
        double r433887 = r433874 * r433882;
        double r433888 = r433887 / r433883;
        double r433889 = r433874 - r433888;
        double r433890 = r433880 ? r433886 : r433889;
        return r433890;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.1
Herbie	2.7

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

Derivation

1. Split input into 2 regimes

2. if x < -1.1645902045388707e-122 or 1.5039508245390077e-300 < x

   1. Initial program 13.4

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

   3. Applied *-un-lft-identity 13.4

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

   4. Applied times-frac 2.3

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

   5. Simplified 2.3

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

   7. Applied *-un-lft-identity 2.3

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

   8. Applied *-un-lft-identity 2.3

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

   9. Applied times-frac 2.3

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

   10. Simplified 2.3

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

   11. Simplified 2.3

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

   if -1.1645902045388707e-122 < x < 1.5039508245390077e-300

   1. Initial program 8.1

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

   2. Taylor expanded around 0 4.4

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{y}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.164590204538870698162352482223739848451 \cdot 10^{-122} \lor \neg \left(x \le 1.503950824539007656103296067322360995492 \cdot 10^{-300}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))