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 r808428 = x;
        double r808429 = y;
        double r808430 = z;
        double r808431 = r808429 - r808430;
        double r808432 = r808428 * r808431;
        double r808433 = r808432 / r808429;
        return r808433;
}

↓

double f(double x, double y, double z) {
        double r808434 = y;
        double r808435 = -1.2027044205997761e-68;
        bool r808436 = r808434 <= r808435;
        double r808437 = x;
        double r808438 = z;
        double r808439 = r808434 - r808438;
        double r808440 = r808439 / r808434;
        double r808441 = r808437 * r808440;
        double r808442 = -2.7394302841690367e-229;
        bool r808443 = r808434 <= r808442;
        double r808444 = r808437 / r808434;
        double r808445 = 1.0;
        double r808446 = r808445 / r808439;
        double r808447 = r808444 / r808446;
        double r808448 = r808434 / r808439;
        double r808449 = r808437 / r808448;
        double r808450 = r808443 ? r808447 : r808449;
        double r808451 = r808436 ? r808441 : r808450;
        return r808451;
}

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 +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))