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

Average Error: 0.1 → 0.3

Time: 8.3s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (* (- z (+ x y)) (/ -0.5 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 (z - (x + y)) * (-0.5 / 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 = (z - (x + y)) * ((-0.5d0) / 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 (z - (x + y)) * (-0.5 / t);
}

Python

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

↓

def code(x, y, z, t):
	return (z - (x + y)) * (-0.5 / 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(z - Float64(x + y)) * Float64(-0.5 / 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 = (z - (x + y)) * (-0.5 / 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[(z - N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.3

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

   [Start] 0.1	\[ \frac{\left(x + y\right) - z}{t \cdot 2} \]
   sub-neg [=>] 0.1	\[ \frac{\color{blue}{\left(x + y\right) + \left(-z\right)}}{t \cdot 2} \]
   +-commutative [=>] 0.1	\[ \frac{\color{blue}{\left(-z\right) + \left(x + y\right)}}{t \cdot 2} \]
   neg-sub0 [=>] 0.1	\[ \frac{\color{blue}{\left(0 - z\right)} + \left(x + y\right)}{t \cdot 2} \]
   associate-+l- [=>] 0.1	\[ \frac{\color{blue}{0 - \left(z - \left(x + y\right)\right)}}{t \cdot 2} \]
   sub0-neg [=>] 0.1	\[ \frac{\color{blue}{-\left(z - \left(x + y\right)\right)}}{t \cdot 2} \]
   neg-mul-1 [=>] 0.1	\[ \frac{\color{blue}{-1 \cdot \left(z - \left(x + y\right)\right)}}{t \cdot 2} \]
   *-commutative [=>] 0.1	\[ \frac{\color{blue}{\left(z - \left(x + y\right)\right) \cdot -1}}{t \cdot 2} \]
   times-frac [=>] 0.1	\[ \color{blue}{\frac{z - \left(x + y\right)}{t} \cdot \frac{-1}{2}} \]
   associate-*l/ [=>] 0.0	\[ \color{blue}{\frac{\left(z - \left(x + y\right)\right) \cdot \frac{-1}{2}}{t}} \]
   associate-*r/ [<=] 0.3	\[ \color{blue}{\left(z - \left(x + y\right)\right) \cdot \frac{\frac{-1}{2}}{t}} \]
   metadata-eval [=>] 0.3	\[ \left(z - \left(x + y\right)\right) \cdot \frac{\color{blue}{-0.5}}{t} \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	10.3
Cost	972

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y - z}{t}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \left(\frac{y}{t} + \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	35.8
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\frac{t}{0.5}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-278} \lor \neg \left(x \leq 7.2 \cdot 10^{-240}\right) \land x \leq 5.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3
Error	17.5
Cost	845

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+181} \lor \neg \left(x \leq -3.7 \cdot 10^{+172}\right) \land x \leq -2.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{\frac{t}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 4
Error	10.5
Cost	844

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y - z}{t}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - x\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	10.5
Cost	844

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y - z}{t}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z - x\right)}{t}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-0.5}{t} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 7
Error	35.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{\frac{t}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 8
Error	40.8
Cost	320

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

Reproduce

herbie shell --seed 2023016 
(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)))