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

Average Error: 33.7 → 1.9

Time: 13.8s

Precision: binary64

Cost: 14084

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{t}, \frac{1}{t}, {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+26)
   (fma (* z (/ z t)) (/ 1.0 t) (pow (/ x y) 2.0))
   (+ (/ x (/ y (/ x y))) (/ (/ z t) (/ t z)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+26) {
		tmp = fma((z * (z / t)), (1.0 / t), pow((x / y), 2.0));
	} else {
		tmp = (x / (y / (x / y))) + ((z / t) / (t / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+26)
		tmp = fma(Float64(z * Float64(z / t)), Float64(1.0 / t), (Float64(x / y) ^ 2.0));
	else
		tmp = Float64(Float64(x / Float64(y / Float64(x / y))) + Float64(Float64(z / t) / Float64(t / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e+26], N[(N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	33.7
Target	0.4
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e26

   1. Initial program 24.5

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

   2. Applied egg-rr 1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\frac{t}{z}}, \frac{1}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]

   3. Applied egg-rr 1.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, \frac{1}{t}, {\left(\frac{x}{y}\right)}^{2}\right) \]

   if 5.0000000000000001e26 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 48.9

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

   2. Applied egg-rr 45.0

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

   3. Applied egg-rr 12.1

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

   4. Applied egg-rr 3.3

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

   5. Applied egg-rr 2.9

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{z}{t}, \frac{1}{t}, {\left(\frac{x}{y}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	7752

\[\begin{array}{l} t_1 := \frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{if}\;t \cdot t \leq 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{z}{t \cdot t}, \frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.4
Cost	1992

\[\begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{if}\;t_1 \leq 10^{+300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	4.6
Cost	1992

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;t_1 + \frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]

Alternative 4
Error	4.1
Cost	1608

\[\begin{array}{l} t_1 := \frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{if}\;t \cdot t \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y} + z \cdot \frac{1}{\frac{t}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	4.6
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y} + \frac{1}{t} \cdot \frac{z}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]

Alternative 6
Error	28.6
Cost	1488

\[\begin{array}{l} t_1 := \frac{\frac{z \cdot z}{t}}{t}\\ t_2 := \frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{if}\;z \cdot z \leq 8.191885151714373 \cdot 10^{-282}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot z \leq 5.6 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot z \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot z \leq 7 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	4.0
Cost	1476

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;t_1 + \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{x}{y}}} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]

Alternative 8
Error	17.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{t}}{t}\\ \end{array} \]

Alternative 9
Error	13.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]

Alternative 10
Error	11.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]

Alternative 11
Error	37.8
Cost	448

\[\frac{\frac{x \cdot x}{y}}{y} \]

Alternative 12
Error	30.4
Cost	448

\[\frac{\frac{x}{\frac{y}{x}}}{y} \]

Reproduce

herbie shell --seed 2022310 
(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))))