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

Average Error: 12.5 → 2.2

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -7.915532472055808 \cdot 10^{+99}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;y \leq 1.7414928146654685 \cdot 10^{-279}:\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt{y}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{z}}}{\sqrt{y}}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.915532472055808e+99)
   (- x (* x (/ z y)))
   (if (<= y 1.7414928146654685e-279)
     (+ x (* (* x z) (/ -1.0 y)))
     (-
      x
      (*
       (* x (* (pow (cbrt z) 2.0) (/ (cbrt (pow (cbrt z) 2.0)) (sqrt y))))
       (/ (cbrt (cbrt z)) (sqrt y)))))))

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 tmp;
	if ((y <= -7.915532472055808e+99)) {
		tmp = ((double) (x - ((double) (x * (z / y)))));
	} else {
		double tmp_1;
		if ((y <= 1.7414928146654685e-279)) {
			tmp_1 = ((double) (x + ((double) (((double) (x * z)) * (-1.0 / y)))));
		} else {
			tmp_1 = ((double) (x - ((double) (((double) (x * ((double) (((double) pow(((double) cbrt(z)), 2.0)) * (((double) cbrt(((double) pow(((double) cbrt(z)), 2.0)))) / ((double) sqrt(y))))))) * (((double) cbrt(((double) cbrt(z)))) / ((double) sqrt(y)))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.2
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if y < -7.91553247205580839e99

   1. Initial program Error: 22.8 bits

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

   2. Simplified Error: 0.0 bits

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

   if -7.91553247205580839e99 < y < 1.74149281466546852e-279

   1. Initial program Error: 6.5 bits

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

   2. Simplified Error: 6.2 bits

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

   4. Applied div-inv Error: 6.2 bits

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

   5. Applied associate-*r* Error: 3.5 bits

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

   if 1.74149281466546852e-279 < y

   1. Initial program Error: 12.0 bits

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

   2. Simplified Error: 2.8 bits

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

   4. Applied *-un-lft-identity Error: 2.8 bits

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

   5. Applied add-cube-cbrt Error: 3.2 bits

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

   6. Applied times-frac Error: 3.2 bits

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

   7. Applied associate-*r* Error: 3.2 bits

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

   8. Simplified Error: 3.2 bits

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

   10. Applied add-sqr-sqrt Error: 3.2 bits

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

   11. Applied add-cube-cbrt Error: 3.2 bits

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

   12. Applied cbrt-prod Error: 3.3 bits

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

   13. Applied times-frac Error: 3.3 bits

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

   14. Applied associate-*r* Error: 2.9 bits

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

   15. Simplified Error: 2.2 bits

       \[\leadsto x - \color{blue}{\left(x \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt{y}}\right)\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{z}}}{\sqrt{y}}\]
3. Recombined 3 regimes into one program.

4. Final simplification Error: 2.2 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.915532472055808 \cdot 10^{+99}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;y \leq 1.7414928146654685 \cdot 10^{-279}:\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt{y}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{z}}}{\sqrt{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))