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

Average Error: 12.2 → 0.6

Time: 11.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \le -7.20588972706695579 \cdot 10^{299}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le -1.225044536923376 \cdot 10^{36}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 1.07630945857270822 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 1.6026833556464087 \cdot 10^{274}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r523803 = x;
        double r523804 = y;
        double r523805 = z;
        double r523806 = r523804 - r523805;
        double r523807 = r523803 * r523806;
        double r523808 = r523807 / r523804;
        return r523808;
}

↓

double f(double x, double y, double z) {
        double r523809 = x;
        double r523810 = y;
        double r523811 = z;
        double r523812 = r523810 - r523811;
        double r523813 = r523809 * r523812;
        double r523814 = r523813 / r523810;
        double r523815 = -7.205889727066956e+299;
        bool r523816 = r523814 <= r523815;
        double r523817 = r523812 / r523810;
        double r523818 = r523809 * r523817;
        double r523819 = -1.225044536923376e+36;
        bool r523820 = r523814 <= r523819;
        double r523821 = 1.0763094585727082e-66;
        bool r523822 = r523814 <= r523821;
        double r523823 = r523810 / r523812;
        double r523824 = r523809 / r523823;
        double r523825 = 1.6026833556464087e+274;
        bool r523826 = r523814 <= r523825;
        double r523827 = r523826 ? r523814 : r523818;
        double r523828 = r523822 ? r523824 : r523827;
        double r523829 = r523820 ? r523814 : r523828;
        double r523830 = r523816 ? r523818 : r523829;
        return r523830;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	2.9
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (* x (- y z)) y) < -7.205889727066956e+299 or 1.6026833556464087e+274 < (/ (* x (- y z)) y)

   1. Initial program 55.4

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

   3. Applied *-un-lft-identity 55.4

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

   4. Applied times-frac 2.3

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

   5. Simplified 2.3

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

   if -7.205889727066956e+299 < (/ (* x (- y z)) y) < -1.225044536923376e+36 or 1.0763094585727082e-66 < (/ (* x (- y z)) y) < 1.6026833556464087e+274

   1. Initial program 0.2

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

   if -1.225044536923376e+36 < (/ (* x (- y z)) y) < 1.0763094585727082e-66

   1. Initial program 7.1

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

   3. Applied associate-/l* 0.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \le -7.20588972706695579 \cdot 10^{299}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le -1.225044536923376 \cdot 10^{36}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 1.07630945857270822 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \le 1.6026833556464087 \cdot 10^{274}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"

  :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))