Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Average Error: 33.9 → 1.0

Time: 7.3s

Precision: binary64

Math

\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \le 2.04599685349181801 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{elif}\;y \cdot y \le 1.3962320537088231 \cdot 10^{307}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (y * y)) <= 2.045996853491818e-307)) {
		VAR = ((double) (((double) (((double) (x / y)) * ((double) (x / y)))) + ((double) (z * ((double) (((double) (z / t)) / t))))));
	} else {
		double VAR_1;
		if ((((double) (y * y)) <= 1.396232053708823e+307)) {
			VAR_1 = ((double) (((double) (x * ((double) (((double) (x / y)) / y)))) + ((double) (((double) (z / t)) * ((double) (z / t))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (x / y)) / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) (x / ((double) cbrt(y)))))) + ((double) (((double) (z / t)) * ((double) (z / t))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.9
Target	0.4
Herbie	1.0

\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

1. Split input into 3 regimes

2. if (* y y) < 2.04599685349181801e-307

   1. Initial program 62.7

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
   2. Using strategy rm

   3. Applied times-frac 61.8

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}}\]
   4. Using strategy rm

   5. Applied times-frac 0.5

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 0.5

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

   8. Applied add-sqr-sqrt 33.8

      \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{1 \cdot t}\]

   9. Applied times-frac 33.8

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

   10. Applied *-un-lft-identity 33.8

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

   11. Applied add-sqr-sqrt 33.8

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

   12. Applied times-frac 33.8

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

   13. Applied swap-sqr 35.3

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

   14. Simplified 35.3

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

   15. Simplified 3.7

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

   if 2.04599685349181801e-307 < (* y y) < 1.3962320537088231e307

   1. Initial program 25.4

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
   2. Using strategy rm

   3. Applied times-frac 7.4

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}}\]
   4. Using strategy rm

   5. Applied times-frac 0.4

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 0.4

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

   8. Applied add-sqr-sqrt 32.6

      \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1 \cdot y} + \frac{z}{t} \cdot \frac{z}{t}\]

   9. Applied times-frac 32.6

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

   10. Applied *-un-lft-identity 32.6

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

   11. Applied add-sqr-sqrt 32.6

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

   12. Applied times-frac 32.7

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

   13. Applied swap-sqr 32.7

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

   14. Simplified 32.6

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

   15. Simplified 0.4

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

   if 1.3962320537088231e307 < (* y y)

   1. Initial program 35.9

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
   2. Using strategy rm

   3. Applied times-frac 20.7

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}}\]
   4. Using strategy rm

   5. Applied times-frac 0.4

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.6

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

   8. Applied *-un-lft-identity 0.6

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

   9. Applied times-frac 0.6

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

   10. Applied associate-*r* 1.1

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

   11. Simplified 1.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \le 2.04599685349181801 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{elif}\;y \cdot y \le 1.3962320537088231 \cdot 10^{307}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))