Logistic function

Average Error: 0.19% → 0.21%

Time: 12.9s

Precision: binary32

Cost: 9792

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

Math

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

↓

\[\frac{1}{e^{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}} \]

FPCore

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

↓

(FPCore (x s) :precision binary32 (/ 1.0 (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 1.0f / 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 Float32(Float32(1.0) / exp(log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 0.19

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

2. Applied egg-rr 0.21

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

3. Applied egg-rr 0.2

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

4. Simplified 0.21

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

   [Start] 0.2	\[ \frac{1}{\frac{1}{e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)}}} \]
   rec-exp [=>] 0.21	\[ \frac{1}{\color{blue}{e^{-\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)}}} \]
   remove-double-neg [=>] 0.21	\[ \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)}}} \]
   distribute-neg-frac [=>] 0.21	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{-x}{s}}}\right)}} \]

5. Final simplification 0.21

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

Alternatives

Alternative 1
Error	0.19%
Cost	3456

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

Alternative 2
Error	9.84%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) - x \cdot \left(x \cdot \frac{-0.5}{s \cdot s}\right)}\\ \end{array} \]

Alternative 3
Error	35.86%
Cost	516

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \frac{\frac{0.5}{s}}{s}\right)}\\ \end{array} \]

Alternative 4
Error	9.67%
Cost	516

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \frac{\frac{0.5}{s}}{s}\right)}\\ \end{array} \]

Alternative 5
Error	38.41%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 6
Error	39.64%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x}{s \cdot \frac{s}{x}}}\\ \end{array} \]

Alternative 7
Error	37.94%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 8
Error	50.85%
Cost	388

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 9
Error	52.25%
Cost	356

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 10
Error	53.6%
Cost	164

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Error	64.82%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))