Logistic function

Average Error: 0.1 → 0.1

Time: 28.4s

Precision: binary32

Cost: 6720

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

Math

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

↓

\[\frac{1}{1 + {\left(e^{x \cdot \frac{-0.5}{s}}\right)}^{2}} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

↓

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp (* x (/ -0.5 s))) 2.0))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

↓

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf((x * (-0.5f / s))), 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((-x / s)))
end function

↓

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((x * ((-0.5e0) / s))) ** 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

↓

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(x * Float32(Float32(-0.5) / s))) ^ Float32(2.0))))
end

MATLAB

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

↓

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp((x * (single(-0.5) / s))) ^ single(2.0)));
end

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

   \[\leadsto \frac{1}{1 + \color{blue}{e^{-0.5 \cdot \frac{x}{s}} \cdot e^{-0.5 \cdot \frac{x}{s}}}} \]

3. Taylor expanded in x around inf 0.1

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{2}}} \]

4. Simplified 0.1

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{x \cdot \frac{-0.5}{s}}\right)}^{2}}} \]
   Proof

   [Start] 0.1	\[ \frac{1}{1 + {\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{2}} \]
   rational_best-simplify-55 [=>] 0.1	\[ \frac{1}{1 + {\left(e^{\color{blue}{x \cdot \frac{-0.5}{s}}}\right)}^{2}} \]

5. Final simplification 0.1

   \[\leadsto \frac{1}{1 + {\left(e^{x \cdot \frac{-0.5}{s}}\right)}^{2}} \]

Alternatives

Alternative 1
Error	0.1
Cost	3456

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

Alternative 2
Error	2.6
Cost	616

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;-1 + \left(1 - \frac{s}{x}\right)\\ \mathbf{elif}\;x \leq -2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{\frac{\frac{2}{s} - \frac{x}{s \cdot s}}{\frac{1}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{2 + \left(\frac{4}{1 + \frac{x}{s}} - -2\right)}\\ \end{array} \]

Alternative 3
Error	3.5
Cost	484

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(-1 - \frac{s}{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{2 + \left(\frac{4}{1 + \frac{x}{s}} - -2\right)}\\ \end{array} \]

Alternative 4
Error	3.5
Cost	420

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(-1 - \frac{s}{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	360

\[\begin{array}{l} t_0 := 1 - \frac{s}{x}\\ \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;-1 + t_0\\ \mathbf{elif}\;x \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	4.9
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(-1 - \frac{s}{x}\right) + 1\\ \mathbf{elif}\;x \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 7
Error	4.9
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(-1 - \frac{s}{x}\right) + 1\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s}{x} + 1}\\ \end{array} \]

Alternative 8
Error	10.4
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{elif}\;x \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 10
Error	5.2
Cost	296

\[\begin{array}{l} t_0 := 1 - \frac{s}{x}\\ \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;-1 + t_0\\ \mathbf{elif}\;x \leq 1.999999967550318 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	17.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12
Error	20.8
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))