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

Average Error: 33.6 → 0.4

Time: 9.5s

Precision: binary64

Cost: 960

Math

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

↓

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

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) (/ 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) {
	return ((x / y) * (x / y)) + ((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) * (x / y)) + ((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) * (x / y)) + ((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) * (x / y)) + ((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(x / y)) + 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) * (x / y)) + ((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[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	33.6
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 33.6

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t}} \]
   Proof
   (+.f64 (*.f64 (/.f64 x y) (/.f64 x y)) (*.f64 (/.f64 z t) (/.f64 z t))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x x) (*.f64 y y))) (*.f64 (/.f64 z t) (/.f64 z t))): 78 points increase in error, 15 points decrease in error
   (+.f64 (/.f64 (*.f64 x x) (*.f64 y y)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 z z) (*.f64 t t)))): 76 points increase in error, 12 points decrease in error

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	10.1
Cost	1992

\[\begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t_1 \leq 10^{-213}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t_1 \leq 5000000000:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t} + x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \end{array} \]

Alternative 2
Error	11.5
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Error	11.5
Cost	964

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	960

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

Alternative 5
Error	37.3
Cost	448

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

Alternative 6
Error	26.1
Cost	448

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

Reproduce

herbie shell --seed 2022325 
(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))))