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

Average Accuracy: 99.9% → 99.9%

Time: 9.2s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 99.9%

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

3. Simplified 99.9%

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

   [Start] 99.9	\[ 0.5 \cdot \frac{y - z}{t} + 0.5 \cdot \frac{x}{t} \]
   +-commutative [=>] 99.9	\[ \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   distribute-lft-out [=>] 99.9	\[ \color{blue}{0.5 \cdot \left(\frac{x}{t} + \frac{y - z}{t}\right)} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	52.0%
Cost	1640

\[\begin{array}{l} t_1 := z \cdot \frac{-0.5}{t}\\ t_2 := 0.5 \cdot \frac{y}{t}\\ t_3 := x \cdot \frac{0.5}{t}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-133}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-271}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8.1 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	52.1%
Cost	1640

\[\begin{array}{l} t_1 := -0.5 \cdot \frac{z}{t}\\ t_2 := 0.5 \cdot \frac{y}{t}\\ t_3 := x \cdot \frac{0.5}{t}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-272}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	52.1%
Cost	1640

\[\begin{array}{l} t_1 := -0.5 \cdot \frac{z}{t}\\ t_2 := 0.5 \cdot \frac{y}{t}\\ t_3 := x \cdot \frac{0.5}{t}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -2.55 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-272}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-131}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	52.2%
Cost	1640

\[\begin{array}{l} t_1 := -0.5 \cdot \frac{z}{t}\\ t_2 := 0.5 \cdot \frac{y}{t}\\ t_3 := \frac{0.5 \cdot x}{t}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{+119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-272}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-20}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	85.3%
Cost	845

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-6} \lor \neg \left(y \leq 1.3 \cdot 10^{+41}\right) \land y \leq 1.65 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 6
Accuracy	89.6%
Cost	845

\[\begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+31} \lor \neg \left(y \leq 1.85 \cdot 10^{+52}\right):\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 7
Accuracy	89.6%
Cost	845

\[\begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{t} - \frac{z}{t}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+31} \lor \neg \left(y \leq 1.4 \cdot 10^{+52}\right):\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 8
Accuracy	52.6%
Cost	717

\[\begin{array}{l} \mathbf{if}\;y \leq 0.48 \lor \neg \left(y \leq 3.9 \cdot 10^{+40}\right) \land y \leq 3.7 \cdot 10^{+96}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 9
Accuracy	78.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+135} \lor \neg \left(z \leq 0.0275\right):\\ \;\;\;\;-0.5 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 10
Accuracy	99.6%
Cost	576

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

Alternative 11
Accuracy	99.9%
Cost	576

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

Alternative 12
Accuracy	36.4%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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)))