Logistic function

Average Error: 0.0 → 0.0

Time: 1.8min

Precision: binary64

Cost: 13312

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

↓

\[\frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}} \]

FPCore

(FPCore (x s) :precision binary64 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

↓

(FPCore (x s) :precision binary64 (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ x s)))))

C

double code(double x, double s) {
	return 1.0 / (1.0 + exp((-x / s)));
}

↓

double code(double x, double s) {
	return 1.0 / (1.0 + pow(exp(-1.0), (x / s)));
}

Fortran

real(8) function code(x, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: s
    code = 1.0d0 / (1.0d0 + exp((-x / s)))
end function

↓

real(8) function code(x, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: s
    code = 1.0d0 / (1.0d0 + (exp((-1.0d0)) ** (x / s)))
end function

Java

public static double code(double x, double s) {
	return 1.0 / (1.0 + Math.exp((-x / s)));
}

↓

public static double code(double x, double s) {
	return 1.0 / (1.0 + Math.pow(Math.exp(-1.0), (x / s)));
}

Python

def code(x, s):
	return 1.0 / (1.0 + math.exp((-x / s)))

↓

def code(x, s):
	return 1.0 / (1.0 + math.pow(math.exp(-1.0), (x / s)))

Julia

function code(x, s)
	return Float64(1.0 / Float64(1.0 + exp(Float64(Float64(-x) / s))))
end

↓

function code(x, s)
	return Float64(1.0 / Float64(1.0 + (exp(-1.0) ^ Float64(x / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = 1.0 / (1.0 + exp((-x / s)));
end

↓

function tmp = code(x, s)
	tmp = 1.0 / (1.0 + (exp(-1.0) ^ (x / s)));
end

Wolfram

code[x_, s_] := N[(1.0 / N[(1.0 + N[Exp[N[((-x) / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, s_] := N[(1.0 / N[(1.0 + N[Power[N[Exp[-1.0], $MachinePrecision], N[(x / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

2. Applied egg-rr 0.0

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}}} \]

Alternatives

Alternative 1
Error	1.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;-x \leq -5 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{1 - -1}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	6976

\[\frac{1}{1 + \frac{1}{e^{\frac{x}{s}}}} \]

Alternative 3
Error	0.0
Cost	6912

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

Alternative 4
Error	3.8
Cost	2252

\[\begin{array}{l} t_0 := \frac{s - x}{s}\\ \mathbf{if}\;-x \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 10^{-60}:\\ \;\;\;\;\frac{1}{\begin{array}{l} \mathbf{if}\;t_0 \ne 0:\\ \;\;\;\;t_0 \cdot \left(1 + \left(\left(1 + \frac{x \cdot x}{s \cdot s}\right) + \frac{x}{s}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{x}{s}\\ \end{array}}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(3 + \frac{1}{\frac{x}{s}}\right)\\ \end{array} \]

Alternative 5
Error	3.8
Cost	1480

\[\begin{array}{l} \mathbf{if}\;-x \leq 5 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 10^{-60}:\\ \;\;\;\;\frac{1}{1 + \left(\left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(3 + \frac{1}{\frac{x}{s}}\right)\\ \end{array} \]

Alternative 6
Error	8.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;-x \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;-x \leq 10^{-172}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(3 + \frac{1}{\frac{x}{s}}\right)\\ \end{array} \]

Alternative 7
Error	4.8
Cost	900

\[\begin{array}{l} \mathbf{if}\;-x \leq 10^{-172}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(3 + \frac{1}{\frac{x}{s}}\right)\\ \end{array} \]

Alternative 8
Error	19.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;-x \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;-x \leq 4 \cdot 10^{-26}:\\ \;\;\;\;0.5 + \frac{0.25 \cdot x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 9
Error	18.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;-x \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;-x \leq 4 \cdot 10^{-26}:\\ \;\;\;\;0.5 + \frac{0.25 \cdot x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 10
Error	19.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;-x \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x - s}{x}\\ \mathbf{elif}\;-x \leq 4 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 11
Error	18.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 12
Error	35.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 13
Error	35.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14
Error	43.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x s)
  :name "Logistic function"
  :precision binary64
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))