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

Average Error: 33.2 → 4.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.71917774278507242 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{1}{y} \cdot \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\left|x\right|}{y \cdot \frac{y}{\left|x\right|}}\right)\\ \end{array}\]

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 ((t <= -6.7191777427850724e-161)) {
		VAR = ((double) fma(((double) (z / t)), ((double) (z / t)), ((double) (((double) (1.0 / y)) * ((double) (x / ((double) (y / x))))))));
	} else {
		VAR = ((double) fma(((double) (z / t)), ((double) (z / t)), ((double) (((double) fabs(x)) / ((double) (y * ((double) (y / ((double) fabs(x))))))))));
	}
	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	4.1

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

Derivation

1. Split input into 2 regimes

2. if t < -6.7191777427850724e-161

   1. Initial program 28.4

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

   2. Simplified 18.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 18.9

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

   5. Applied times-frac 13.5

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

   6. Simplified 4.2

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

   if -6.7191777427850724e-161 < t

   1. Initial program 36.9

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

   2. Simplified 18.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 18.6

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

   5. Applied add-sqr-sqrt 18.6

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

   6. Applied times-frac 18.6

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

   7. Simplified 18.6

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\left|x\right|} \cdot \frac{\sqrt{x \cdot x}}{y \cdot y}\right)\]

   8. Simplified 4.3

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|x\right| \cdot \color{blue}{\frac{\frac{\left|x\right|}{y}}{y}}\right)\]
   9. Using strategy rm

   10. Applied clear-num 4.4

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|x\right| \cdot \frac{\color{blue}{\frac{1}{\frac{y}{\left|x\right|}}}}{y}\right)\]

   11. Applied associate-/l/ 4.5

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left|x\right| \cdot \color{blue}{\frac{1}{y \cdot \frac{y}{\left|x\right|}}}\right)\]

   12. Applied un-div-inv 4.1

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{\left|x\right|}{y \cdot \frac{y}{\left|x\right|}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.71917774278507242 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{1}{y} \cdot \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\left|x\right|}{y \cdot \frac{y}{\left|x\right|}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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) (pow (/ z t) 2))

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