Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B

Average Error: 0.1 → 0.1

Time: 8.3s

Precision: binary64

Cost: 704

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\left(\frac{x - z}{t} + \frac{y}{t}\right) \cdot 0.5 \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

↓

(FPCore (x y z t) :precision binary64 (* (+ (/ (- x z) t) (/ y t)) 0.5))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

↓

double code(double x, double y, double z, double t) {
	return (((x - z) / t) + (y / t)) * 0.5;
}

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 + y) - z) / (t * 2.0d0)
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 - z) / t) + (y / t)) * 0.5d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

↓

public static double code(double x, double y, double z, double t) {
	return (((x - z) / t) + (y / t)) * 0.5;
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

↓

def code(x, y, z, t):
	return (((x - z) / t) + (y / t)) * 0.5

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - z) / t) + Float64(y / t)) * 0.5)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

↓

function tmp = code(x, y, z, t)
	tmp = (((x - z) / t) + (y / t)) * 0.5;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision] + N[(y / t), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in y around 0 0.1

   \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t} + 0.5 \cdot \frac{x - z}{t}} \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(\frac{x - z}{t} + \frac{y}{t}\right) \cdot 0.5} \]

4. Final simplification 0.1

   \[\leadsto \left(\frac{x - z}{t} + \frac{y}{t}\right) \cdot 0.5 \]

Alternatives

Alternative 1
Error	27.0
Cost	1376

\[\begin{array}{l} t_1 := \frac{y \cdot 0.5}{t}\\ t_2 := \frac{x}{\frac{t}{0.5}}\\ t_3 := -0.5 \cdot \frac{z}{t}\\ \mathbf{if}\;x \leq -1.0422830064006472 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.490185412162121 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -0.04818218621818409:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.963724248589354 \cdot 10^{-183}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.6232878753408387 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5710474559422605 \cdot 10^{-291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 4.539652451897301 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.969706393242968 \cdot 10^{-71}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.0
Cost	976

\[\begin{array}{l} t_1 := -0.5 \cdot \frac{z}{t}\\ t_2 := \frac{x + y}{t \cdot 2}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.9731126194282375 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.8411463389161615 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.7045725971702525 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.4
Cost	844

\[\begin{array}{l} t_1 := \frac{x - z}{t} \cdot 0.5\\ \mathbf{if}\;x \leq -1 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -23121653.143168956:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{elif}\;x \leq -7.649831640126826 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	712

\[\begin{array}{l} t_1 := \frac{x - z}{t} \cdot 0.5\\ \mathbf{if}\;z \leq -1.8411463389161615 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9456849078039833 \cdot 10^{+43}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.3
Cost	576

\[\left(x + \left(y - z\right)\right) \cdot \frac{0.5}{t} \]

Alternative 6
Error	0.1
Cost	576

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

Alternative 7
Error	29.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -0.04818218621818409:\\ \;\;\;\;\frac{x}{\frac{t}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \end{array} \]

Alternative 8
Error	41.6
Cost	320

\[\frac{x}{\frac{t}{0.5}} \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))