Logistic function

Average Error: 0.1 → 0.1

Time: 8.0s

Precision: binary32

Cost: 6720

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp 2.0) (* -0.5 (/ 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 + powf(expf(2.0f), (-0.5f * (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(2.0e0) ** ((-0.5e0) * (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(2.0)) ^ Float32(Float32(-0.5) * 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(single(2.0)) ^ (single(-0.5) * (x / s))));
end

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	3456

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

Alternative 2
Error	0.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\left(1 + \frac{x}{s}\right) + \frac{x}{s} \cdot \frac{x \cdot 0.5}{s}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	1.3
Cost	516

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	1.3
Cost	360

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	13.7
Cost	100

\[\begin{array}{l} \mathbf{if}\;x \leq 4.999999980020986 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	20.9
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2022339 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))