Logistic function

Average Error: 0.1 → 0.1

Time: 9.5s

Precision: binary32

Cost: 3456

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

Math

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

↓

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

FPCore

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

↓

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

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 + expf((-x / s)));
}

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 / s)))
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(Float32(-x) / s))))
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 / s)));
end

Derivation

1. Initial program 0.1

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

2. Final simplification 0.1

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

Alternatives

Alternative 1
Error	9.9
Cost	3428

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4999999552572828 \cdot 10^{-35}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 2
Error	12.0
Cost	292

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;\left(1 - \frac{s}{x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 3
Error	16.6
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{-\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4
Error	17.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5
Error	20.6
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))