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

Average Error: 12.6 → 3.3

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.202704420599776119000768348402627115529 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;y \le -2.739430284169036711408746407652821381307 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{1}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r681834 = x;
        double r681835 = y;
        double r681836 = z;
        double r681837 = r681835 - r681836;
        double r681838 = r681834 * r681837;
        double r681839 = r681838 / r681835;
        return r681839;
}

↓

double f(double x, double y, double z) {
        double r681840 = y;
        double r681841 = -1.2027044205997761e-68;
        bool r681842 = r681840 <= r681841;
        double r681843 = x;
        double r681844 = z;
        double r681845 = r681840 - r681844;
        double r681846 = r681845 / r681840;
        double r681847 = r681843 * r681846;
        double r681848 = -2.7394302841690367e-229;
        bool r681849 = r681840 <= r681848;
        double r681850 = r681843 / r681840;
        double r681851 = 1.0;
        double r681852 = r681851 / r681845;
        double r681853 = r681850 / r681852;
        double r681854 = r681840 / r681845;
        double r681855 = r681843 / r681854;
        double r681856 = r681849 ? r681853 : r681855;
        double r681857 = r681842 ? r681847 : r681856;
        return r681857;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.6
Target	3.1
Herbie	3.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1.2027044205997761e-68

   1. Initial program 15.0

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

   3. Applied *-un-lft-identity 15.0

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

   4. Applied times-frac 0.4

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

   5. Simplified 0.4

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

   if -1.2027044205997761e-68 < y < -2.7394302841690367e-229

   1. Initial program 7.4

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

   3. Applied associate-/l* 6.9

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

   5. Applied div-inv 7.0

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

   6. Applied associate-/r* 8.4

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

   if -2.7394302841690367e-229 < y

   1. Initial program 12.2

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

   3. Applied associate-/l* 4.0

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.202704420599776119000768348402627115529 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;y \le -2.739430284169036711408746407652821381307 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{1}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))