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

Average Error: 33.5 → 0.4

Time: 11.1s

Precision: binary64

Cost: 1216

Math

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

↓

\[\frac{1}{\frac{1}{\frac{\frac{x}{y}}{\frac{y}{x}}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (+ (/ 1.0 (/ 1.0 (/ (/ 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 (1.0 / (1.0 / ((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 = (1.0d0 / (1.0d0 / ((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 (1.0 / (1.0 / ((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 (1.0 / (1.0 / ((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(1.0 / Float64(1.0 / 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 = (1.0 / (1.0 / ((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[(1.0 / N[(1.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	33.5
Target	0.4
Herbie	0.4

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

Derivation

1. Initial program 33.5

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

2. Simplified 0.4

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

   [Start] 33.5	\[ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   times-frac [=>] 19.5	\[ \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]
   times-frac [=>] 0.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	21.1
Cost	1608

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 1.06 \cdot 10^{-243}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 2
Error	21.1
Cost	1481

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 5.2 \cdot 10^{-243} \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.3
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t} + x \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	960

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

Alternative 5
Error	0.4
Cost	960

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

Alternative 6
Error	0.4
Cost	960

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

Alternative 7
Error	26.9
Cost	448

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

Reproduce

herbie shell --seed 2023039 
(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))))