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

Average Error: 0.1 → 0.1

Time: 14.1s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

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

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 0.5 * ((y / t) + ((x - z) / t));
}

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 = 0.5d0 * ((y / t) + ((x - z) / t))
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 0.5 * ((y / t) + ((x - z) / t));
}

Python

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

↓

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

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(0.5 * Float64(Float64(y / t) + Float64(Float64(x - z) / t)))
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 = 0.5 * ((y / t) + ((x - z) / t));
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[(0.5 * N[(N[(y / t), $MachinePrecision] + N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $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. Simplified 0.1

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

   [Start] 0.1	\[ 0.5 \cdot \frac{y}{t} + 0.5 \cdot \frac{x - z}{t} \]
   rational.json-simplify-1 [=>] 0.1	\[ \color{blue}{0.5 \cdot \frac{x - z}{t} + 0.5 \cdot \frac{y}{t}} \]
   rational.json-simplify-2 [=>] 0.1	\[ 0.5 \cdot \frac{x - z}{t} + \color{blue}{\frac{y}{t} \cdot 0.5} \]
   rational.json-simplify-51 [=>] 0.1	\[ \color{blue}{0.5 \cdot \left(\frac{y}{t} + \frac{x - z}{t}\right)} \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	8.7
Cost	1236

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{x - z}{t}\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \mathbf{elif}\;y \leq 430000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+58}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{y}{t} - \frac{z}{t}\right)\\ \end{array} \]

Alternative 2
Error	8.8
Cost	1108

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y - z}{t}\\ t_2 := 0.5 \cdot \frac{x - z}{t}\\ \mathbf{if}\;y \leq 3.15 \cdot 10^{-90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 500000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \frac{y + x}{t}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.4
Cost	976

\[\begin{array}{l} t_1 := \frac{z}{t} \cdot -0.5\\ t_2 := 0.5 \cdot \frac{y + x}{t}\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	26.5
Cost	848

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y}{t}\\ t_2 := \frac{-0.5}{t} \cdot z\\ \mathbf{if}\;x \leq -3 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-191}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	26.4
Cost	848

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y}{t}\\ t_2 := \frac{z}{t} \cdot -0.5\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.62 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.7
Cost	844

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y + x}{t}\\ t_2 := 0.5 \cdot \frac{x - z}{t}\\ \mathbf{if}\;y \leq 460000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.3
Cost	576

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

Alternative 8
Error	0.1
Cost	576

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

Alternative 9
Error	27.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+14}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 10
Error	41.4
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(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)))