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

Average Accuracy: 47.6% → 99.5%

Time: 12.7s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (+ (/ (/ x y) (/ y x)) (/ (/ 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) {
	return ((x / y) / (y / x)) + ((z / t) / (t / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) / (y / x)) + ((z / t) / (t / z))
end function

Java

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

↓

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

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

↓

def code(x, y, z, t):
	return ((x / y) / (y / x)) + ((z / t) / (t / z))

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)
	return Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(z / t) / Float64(t / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

↓

function tmp = code(x, y, z, t)
	tmp = ((x / y) / (y / x)) + ((z / t) / (t / z));
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_] := N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	47.6%
Target	99.3%
Herbie	99.5%

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

Derivation

1. Initial program 47.6%

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

2. Simplified 99.3%

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

   [Start] 47.6	\[ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   times-frac [=>] 71.0	\[ \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]
   times-frac [=>] 99.3	\[ \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

3. Applied egg-rr 99.4%

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

   [Start] 99.3	\[ \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]
   associate-*r/ [=>] 92.7	\[ \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   associate-/l* [=>] 99.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

4. Applied egg-rr 99.5%

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

   [Start] 99.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   clear-num [=>] 99.5	\[ \frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   un-div-inv [=>] 99.5	\[ \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	66.5%
Cost	1481

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 152000 \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	95.2%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-150} \lor \neg \left(y \leq 1.6 \cdot 10^{-216}\right):\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}} + x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	66.8%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{-288}:\\ \;\;\;\;\frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	960

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

Alternative 5
Accuracy	99.4%
Cost	960

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

Alternative 6
Accuracy	59.0%
Cost	448

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

Reproduce

herbie shell --seed 2023137 
(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))))