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

Average Accuracy: 47.6% → 99.5%

Time: 14.8s

Precision: binary64

Cost: 1152

Math

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

↓

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

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 y) x) (/ 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 / y) * x) / (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) / y) * x) / (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 / y) * x) / (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 / y) * x) / (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(Float64(-1.0 / y) * x) / Float64(y / Float64(-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 / y) * x) / (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[(N[(-1.0 / y), $MachinePrecision] * x), $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 [=>] 70.0	\[ \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]
   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	\[ \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{z}{t} \]
   clear-num [=>] 99.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{1}{\frac{t}{z}}} \]
   un-div-inv [=>] 99.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

4. Applied egg-rr 99.2%

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

   [Start] 99.4	\[ \frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   associate-*r/ [=>] 93.7	\[ \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   associate-/l* [=>] 99.5	\[ \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   clear-num [=>] 99.4	\[ \frac{\color{blue}{\frac{1}{\frac{y}{x}}}}{\frac{y}{x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   frac-2neg [=>] 99.4	\[ \frac{\color{blue}{\frac{-1}{-\frac{y}{x}}}}{\frac{y}{x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   associate-/l/ [=>] 99.2	\[ \color{blue}{\frac{-1}{\frac{y}{x} \cdot \left(-\frac{y}{x}\right)}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   metadata-eval [=>] 99.2	\[ \frac{\color{blue}{-1}}{\frac{y}{x} \cdot \left(-\frac{y}{x}\right)} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   frac-2neg [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \left(-\color{blue}{\frac{-y}{-x}}\right)} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   distribute-neg-frac [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \color{blue}{\frac{-\left(-y\right)}{-x}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   *-un-lft-identity [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{-\left(-\color{blue}{1 \cdot y}\right)}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   distribute-lft-neg-in [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{-\color{blue}{\left(-1\right) \cdot y}}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   distribute-lft-neg-in [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{\color{blue}{\left(-\left(-1\right)\right) \cdot y}}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   metadata-eval [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{\left(-\color{blue}{-1}\right) \cdot y}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   metadata-eval [=>] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{\color{blue}{1} \cdot y}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   *-un-lft-identity [<=] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{\color{blue}{y}}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]

5. Simplified 99.5%

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

   [Start] 99.2	\[ \frac{-1}{\frac{y}{x} \cdot \frac{y}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   associate-/r* [=>] 99.4	\[ \color{blue}{\frac{\frac{-1}{\frac{y}{x}}}{\frac{y}{-x}}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]
   associate-/r/ [=>] 99.5	\[ \frac{\color{blue}{\frac{-1}{y} \cdot x}}{\frac{y}{-x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \]

6. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	68.2%
Cost	1672

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 2.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y} \cdot x}{\frac{y}{-x}}\\ \end{array} \]

Alternative 2
Accuracy	68.1%
Cost	1481

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 1.5 \cdot 10^{-160} \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	68.2%
Cost	1480

\[\begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t_1 \leq 3.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	86.2%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 4 \cdot 10^{+189}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t} + x \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \end{array} \]

Alternative 5
Accuracy	87.5%
Cost	960

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

Alternative 6
Accuracy	99.3%
Cost	960

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

Alternative 7
Accuracy	99.4%
Cost	960

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

Alternative 8
Accuracy	99.4%
Cost	960

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

Alternative 9
Accuracy	58.5%
Cost	448

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

Reproduce

herbie shell --seed 2023138 
(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))))