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

Average Error: 0.1 → 0.1

Time: 8.5s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (+ (* 0.5 (/ (- y z) t)) (* 0.5 (/ x 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 - z) / t)) + (0.5 * (x / 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 - z) / t)) + (0.5d0 * (x / 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 - z) / t)) + (0.5 * (x / 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 - z) / t)) + (0.5 * (x / 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(Float64(0.5 * Float64(Float64(y - z) / t)) + Float64(0.5 * Float64(x / 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 - z) / t)) + (0.5 * (x / 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[(N[(0.5 * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	26.4
Cost	1244

\[\begin{array}{l} t_1 := z \cdot \frac{-0.5}{t}\\ t_2 := y \cdot \frac{0.5}{t}\\ \mathbf{if}\;x \leq -2.7459369957840644 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -6.526256346597655 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.741720364843608 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2384659789149964 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.337777309831154 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.4456307688211523 \cdot 10^{-302}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5139684045868965 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	26.4
Cost	1244

\[\begin{array}{l} t_1 := z \cdot \frac{-0.5}{t}\\ t_2 := \frac{0.5 \cdot y}{t}\\ \mathbf{if}\;x \leq -2.7459369957840644 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -6.526256346597655 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.741720364843608 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2384659789149964 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.337777309831154 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.4456307688211523 \cdot 10^{-302}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5139684045868965 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	26.3
Cost	1244

\[\begin{array}{l} t_1 := -0.5 \cdot \frac{z}{t}\\ t_2 := \frac{0.5 \cdot y}{t}\\ \mathbf{if}\;x \leq -2.7459369957840644 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -6.526256346597655 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.741720364843608 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2384659789149964 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.337777309831154 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.4456307688211523 \cdot 10^{-302}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.5139684045868965 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	11.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6721304867145286 \cdot 10^{+96}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \]

Alternative 5
Error	11.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6721304867145286 \cdot 10^{+96}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \]

Alternative 6
Error	6.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8883363713880545 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \]

Alternative 7
Error	6.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -6.588970868030848 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y - z}}\\ \end{array} \]

Alternative 8
Error	6.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8883363713880545 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 9
Error	0.3
Cost	576

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

Alternative 10
Error	0.1
Cost	576

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

Alternative 11
Error	27.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7459369957840644 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]

Alternative 12
Error	27.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7459369957840644 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.5}{t}\\ \end{array} \]

Alternative 13
Error	41.0
Cost	320

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

Reproduce

herbie shell --seed 2022295 
(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)))