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

Average Error: 33.4 → 2.2

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5.6706044911068 \cdot 10^{+264}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{1}{t \cdot \frac{t}{z}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((double) (z * z)) <= 5.6706044911068e+264)) {
		VAR = ((double) (((double) ((x / y) * (x / y))) + (((double) (z * (z / t))) / t)));
	} else {
		VAR = ((double) (((double) ((x / y) * (x / y))) + ((double) (z * (1.0 / ((double) (t * (t / z))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	33.4
Target	0.4
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 5.6706044911067997e264

   1. Initial program 28.8

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

   2. Simplified 22.7

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

   4. Applied add-sqr-sqrt 43.1

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

   5. Applied times-frac 39.1

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

   6. Applied add-sqr-sqrt 39.1

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

   7. Applied unswap-sqr 37.2

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

   8. Simplified 37.1

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

   9. Simplified 10.8

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

   11. Applied *-un-lft-identity 10.8

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

   12. Applied times-frac 4.1

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \color{blue}{\left(\frac{1}{t} \cdot \frac{z}{t}\right)}\]
   13. Using strategy rm

   14. Applied associate-*r/ 4.1

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

   15. Applied associate-*r/ 1.7

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

   16. Simplified 1.7

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

   if 5.6706044911067997e264 < (* z z)

   1. Initial program 58.5

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

   2. Simplified 32.1

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

   4. Applied add-sqr-sqrt 47.9

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

   5. Applied times-frac 45.3

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

   6. Applied add-sqr-sqrt 45.3

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

   7. Applied unswap-sqr 43.9

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

   8. Simplified 43.9

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

   9. Simplified 24.1

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

   11. Applied clear-num 24.1

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

   12. Simplified 5.2

       \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{1}{\color{blue}{t \cdot \frac{t}{z}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5.6706044911068 \cdot 10^{+264}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot \frac{z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{1}{t \cdot \frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))