Logistic function

Average Accuracy: 99.8% → 99.9%

Time: 10.8s

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 99.9%

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

2. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{1}{1 + e^{\frac{-x}{s}}} \]
   div-inv [=>] 99.9	\[ \frac{1}{1 + e^{\color{blue}{\left(-x\right) \cdot \frac{1}{s}}}} \]
   exp-prod [=>] 83.0	\[ \frac{1}{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\frac{1}{s}\right)}}} \]
   neg-mul-1 [=>] 83.0	\[ \frac{1}{1 + {\left(e^{\color{blue}{-1 \cdot x}}\right)}^{\left(\frac{1}{s}\right)}} \]
   exp-prod [=>] 83.0	\[ \frac{1}{1 + {\color{blue}{\left({\left(e^{-1}\right)}^{x}\right)}}^{\left(\frac{1}{s}\right)}} \]
   pow-pow [=>] 99.9	\[ \frac{1}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(x \cdot \frac{1}{s}\right)}}} \]
   div-inv [<=] 99.9	\[ \frac{1}{1 + {\left(e^{-1}\right)}^{\color{blue}{\left(\frac{x}{s}\right)}}} \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}} \]
   add-exp-log [=>] 99.9	\[ \color{blue}{e^{\log \left(\frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}}\right)}} \]
   log-rec [=>] 99.8	\[ e^{\color{blue}{-\log \left(1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]
   log1p-udef [<=] 99.9	\[ e^{-\color{blue}{\mathsf{log1p}\left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]
   pow-exp [=>] 99.9	\[ e^{-\mathsf{log1p}\left(\color{blue}{e^{-1 \cdot \frac{x}{s}}}\right)} \]
   associate-*r/ [=>] 99.9	\[ e^{-\mathsf{log1p}\left(e^{\color{blue}{\frac{-1 \cdot x}{s}}}\right)} \]
   neg-mul-1 [<=] 99.9	\[ e^{-\mathsf{log1p}\left(e^{\frac{\color{blue}{-x}}{s}}\right)} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	6752

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

Alternative 2
Accuracy	99.8%
Cost	6656

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

Alternative 3
Accuracy	99.8%
Cost	3456

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

Alternative 4
Accuracy	76.6%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	94.2%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.8%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0 \cdot \frac{s}{x}\right)\\ \end{array} \]

Alternative 7
Accuracy	74.4%
Cost	520

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	70.3%
Cost	360

\[\begin{array}{l} \mathbf{if}\;-x \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;-x \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]

Alternative 9
Accuracy	46.0%
Cost	228

\[\begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10
Accuracy	69.0%
Cost	228

\[\begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	46.0%
Cost	164

\[\begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12
Accuracy	34.8%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))