Logistic function

Average Error: 0.1 → 0.1

Time: 8.6s

Precision: binary32

Cost: 3456

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

Math

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

↓

\[\frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)} \]

FPCore

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

↓

(FPCore (x s) :precision binary32 (/ 1.0 (+ 2.0 (expm1 (/ (- 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 / (2.0f + expm1f((-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 Float32(Float32(1.0) / Float32(Float32(2.0) + expm1(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 0.1

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

4. Applied egg-rr 0.3

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{e^{\frac{x}{s} \cdot -1} + 1}\right)} - 1} \]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)}} \]
   Proof

   [Start] 0.3	\[ e^{\mathsf{log1p}\left(\frac{1}{e^{\frac{x}{s} \cdot -1} + 1}\right)} - 1 \]
   expm1-def [=>] 0.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{e^{\frac{x}{s} \cdot -1} + 1}\right)\right)} \]
   expm1-log1p [=>] 0.1	\[ \color{blue}{\frac{1}{e^{\frac{x}{s} \cdot -1} + 1}} \]
   metadata-eval [<=] 0.1	\[ \frac{1}{e^{\frac{x}{s} \cdot -1} + \color{blue}{\left(-1 + 2\right)}} \]
   associate-+l+ [<=] 0.1	\[ \frac{1}{\color{blue}{\left(e^{\frac{x}{s} \cdot -1} + -1\right) + 2}} \]
   metadata-eval [<=] 0.1	\[ \frac{1}{\left(e^{\frac{x}{s} \cdot -1} + \color{blue}{\left(-1\right)}\right) + 2} \]
   sub-neg [<=] 0.1	\[ \frac{1}{\color{blue}{\left(e^{\frac{x}{s} \cdot -1} - 1\right)} + 2} \]
   +-commutative [<=] 0.1	\[ \frac{1}{\color{blue}{2 + \left(e^{\frac{x}{s} \cdot -1} - 1\right)}} \]
   *-commutative [<=] 0.1	\[ \frac{1}{2 + \left(e^{\color{blue}{-1 \cdot \frac{x}{s}}} - 1\right)} \]
   expm1-def [=>] 0.1	\[ \frac{1}{2 + \color{blue}{\mathsf{expm1}\left(-1 \cdot \frac{x}{s}\right)}} \]
   associate-*r/ [=>] 0.1	\[ \frac{1}{2 + \mathsf{expm1}\left(\color{blue}{\frac{-1 \cdot x}{s}}\right)} \]
   neg-mul-1 [<=] 0.1	\[ \frac{1}{2 + \mathsf{expm1}\left(\frac{\color{blue}{-x}}{s}\right)} \]

6. Final simplification 0.1

   \[\leadsto \frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)} \]

Alternatives

Alternative 1
Error	0.1
Cost	3456

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

Alternative 2
Error	10.0
Cost	196

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

Alternative 3
Error	20.9
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023237 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))