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

Average Error: 12.5 → 1.8

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.639338943466909 \cdot 10^{+72} \lor \neg \left(z \leq 8.55574750243714 \cdot 10^{+144}\right) \land z \leq 1.2735314140732662 \cdot 10^{+299}:\\ \;\;\;\;x - \left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return (((double) (x * ((double) (y - z)))) / y);
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((z <= -2.639338943466909e+72) || (!(z <= 8.55574750243714e+144) && (z <= 1.2735314140732662e+299)))) {
		VAR = ((double) (x - ((double) (((double) ((x / ((double) pow(((double) cbrt(y)), 2.0))) * ((double) (((double) cbrt(z)) * (((double) cbrt(z)) / ((double) cbrt(((double) pow(((double) cbrt(y)), 2.0))))))))) * (((double) cbrt(z)) / ((double) cbrt(((double) cbrt(y)))))))));
	} else {
		VAR = ((double) (x - ((double) (x * (z / y)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.0
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.63933894346690909e72 or 8.55574750243713944e144 < z < 1.2735314140732662e299

   1. Initial program Error: 12.7 bits

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

   2. Simplified Error: 9.5 bits

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

   4. Applied add-cube-cbrt Error: 10.1 bits

      \[\leadsto x - x \cdot \frac{z}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

   5. Applied *-un-lft-identity Error: 10.1 bits

      \[\leadsto x - x \cdot \frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

   6. Applied times-frac Error: 10.1 bits

      \[\leadsto x - x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right)}\]

   7. Applied associate-*r* Error: 6.2 bits

      \[\leadsto x - \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{z}{\sqrt[3]{y}}}\]

   8. Simplified Error: 6.2 bits

      \[\leadsto x - \color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{z}{\sqrt[3]{y}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt Error: 6.3 bits

       \[\leadsto x - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

   11. Applied cbrt-prod Error: 6.3 bits

       \[\leadsto x - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\color{blue}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}\]

   12. Applied add-cube-cbrt Error: 6.5 bits

       \[\leadsto x - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}\]

   13. Applied times-frac Error: 6.5 bits

       \[\leadsto x - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\]

   14. Applied associate-*r* Error: 4.4 bits

       \[\leadsto x - \color{blue}{\left(\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y}}}}\]

   15. Simplified Error: 4.4 bits

       \[\leadsto x - \color{blue}{\left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{2}}}\right)\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y}}}\]

   if -2.63933894346690909e72 < z < 8.55574750243713944e144 or 1.2735314140732662e299 < z

   1. Initial program Error: 12.4 bits

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

   2. Simplified Error: 0.9 bits

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

4. Final simplification Error: 1.8 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.639338943466909 \cdot 10^{+72} \lor \neg \left(z \leq 8.55574750243714 \cdot 10^{+144}\right) \land z \leq 1.2735314140732662 \cdot 10^{+299}:\\ \;\;\;\;x - \left(\frac{x}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))