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

Average Error: 33.2 → 2.1

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.28037951004661684 \cdot 10^{164}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;z \le 6.24132298638735279 \cdot 10^{95}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \left(z \cdot \frac{z}{t}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

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 ((z <= -6.280379510046617e+164)) {
		VAR = ((double) (((double) (x * ((double) (((double) (x / y)) / y)))) + ((double) (((double) (z / t)) * ((double) (z / t))))));
	} else {
		double VAR_1;
		if ((z <= 6.241322986387353e+95)) {
			VAR_1 = ((double) (((double) (((double) (x / y)) * ((double) (x / y)))) + ((double) (((double) (z * ((double) (z / t)))) * ((double) (1.0 / t))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (x / y)))) / 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	33.2
Target	0.4
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if z < -6.28037951004661684e164

   1. Initial program 64.0

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

   3. Applied times-frac 64.0

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

   5. Applied times-frac 0.5

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

   7. Applied *-un-lft-identity 0.5

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

   8. Applied add-sqr-sqrt 31.8

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

   9. Applied times-frac 31.7

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

   10. Applied *-un-lft-identity 31.7

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

   11. Applied add-sqr-sqrt 31.8

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

   12. Applied times-frac 31.8

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

   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 32.7

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

   15. Simplified 3.0

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

   if -6.28037951004661684e164 < z < 6.24132298638735279e95

   1. Initial program 29.0

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

   3. Applied times-frac 13.0

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

   5. Applied times-frac 0.4

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

   7. Applied div-inv 0.4

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

   8. Applied associate-*r* 1.9

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

   9. Simplified 1.9

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

   if 6.24132298638735279e95 < z

   1. Initial program 51.3

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

   3. Applied times-frac 44.7

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

   5. Applied times-frac 0.5

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

   7. Applied associate-*r/ 3.7

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

   8. Simplified 3.7

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.28037951004661684 \cdot 10^{164}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;z \le 6.24132298638735279 \cdot 10^{95}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \left(z \cdot \frac{z}{t}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{y} + \frac{z}{t} \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))))