Logistic function

Average Error: 0.1 → 0.0

Time: 11.4s

Precision: binary32

Cost: 9760

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x s) :precision binary32 (exp (- (log1p (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 expf(-log1pf(expf((-x / s))));
}

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 exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \]
   *-rgt-identity [<=] 0.0	\[ e^{-\color{blue}{\mathsf{log1p}\left(e^{-\frac{x}{s}}\right) \cdot 1}} \]
   *-rgt-identity [=>] 0.0	\[ e^{-\color{blue}{\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)}} \]
   distribute-neg-frac [=>] 0.0	\[ e^{-\mathsf{log1p}\left(e^{\color{blue}{\frac{-x}{s}}}\right)} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.1
Cost	3456

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

Alternative 2
Error	10.7
Cost	388

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

Alternative 3
Error	16.4
Cost	356

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

Alternative 4
Error	17.1
Cost	228

\[\begin{array}{l} \mathbf{if}\;-x \leq 5.9999998100067255 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \]

Alternative 5
Error	16.9
Cost	164

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6
Error	20.7
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023041 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))