Logistic function

Average Error: 0.1 → 0.1

Time: 14.2s

Precision: binary32

Cost: 12992

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

Math

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

↓

\[{\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-2} \]

FPCore

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

↓

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

C

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

↓

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

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 hypot(Float32(1.0), (exp(Float32(x / s)) ^ Float32(-0.5))) ^ 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 = hypot(single(1.0), (exp((x / s)) ^ single(-0.5))) ^ single(-2.0);
end

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 0.3

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

4. Simplified 0.1

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

   [Start] 0.3	\[ {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-1} \]
   pow-sqr [=>] 0.1	\[ \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{\left(2 \cdot -1\right)}} \]
   metadata-eval [=>] 0.1	\[ {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{\color{blue}{-2}} \]

5. Final simplification 0.1

   \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)\right)}^{-2} \]

Alternatives

Alternative 1
Error	0.1
Cost	9824

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

Alternative 2
Error	0.1
Cost	9760

\[e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]

Alternative 3
Error	0.1
Cost	3456

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

Alternative 4
Error	2.0
Cost	836

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

Alternative 5
Error	8.0
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 6
Error	2.3
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 7
Error	2.3
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 8
Error	8.6
Cost	520

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

Alternative 9
Error	2.5
Cost	516

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

Alternative 10
Error	10.7
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 11
Error	17.3
Cost	228

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

Alternative 12
Error	17.3
Cost	164

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

Alternative 13
Error	20.9
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023057 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))