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

Average Error: 12.7 → 2.6

Time: 7.5s

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 r561690 = x;
        double r561691 = y;
        double r561692 = z;
        double r561693 = r561691 - r561692;
        double r561694 = r561690 * r561693;
        double r561695 = r561694 / r561691;
        return r561695;
}

↓

double f(double x, double y, double z) {
        double r561696 = y;
        double r561697 = 4.014228494605548e-282;
        bool r561698 = r561696 <= r561697;
        double r561699 = x;
        double r561700 = z;
        double r561701 = r561696 - r561700;
        double r561702 = r561696 / r561701;
        double r561703 = r561699 / r561702;
        double r561704 = 2.3679111051945716e-120;
        bool r561705 = r561696 <= r561704;
        double r561706 = r561699 * r561700;
        double r561707 = 1.0;
        double r561708 = r561707 / r561696;
        double r561709 = r561706 * r561708;
        double r561710 = r561699 - r561709;
        double r561711 = r561701 / r561696;
        double r561712 = r561699 * r561711;
        double r561713 = r561705 ? r561710 : r561712;
        double r561714 = r561698 ? r561703 : r561713;
        return r561714;
}

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 +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 -2.060202331921739e104) (- x (/ (* z x) y)) (if (< z 1.69397660138285259e213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))