Logistic function

Average Accuracy: 99.8% → 99.8%

Time: 11.3s

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 99.8%

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

2. Applied egg-rr 99.7%

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

   [Start] 99.8	\[ \frac{1}{1 + e^{\frac{-x}{s}}} \]
   distribute-frac-neg [=>] 99.8	\[ \frac{1}{1 + e^{\color{blue}{-\frac{x}{s}}}} \]
   exp-neg [=>] 99.7	\[ \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
   add-sqr-sqrt [=>] 49.8	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}} \]
   sqrt-unprod [=>] 61.2	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x \cdot x}}}{s}}}} \]
   sqr-neg [<=] 61.2	\[ \frac{1}{1 + \frac{1}{e^{\frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{s}}}} \]
   sqrt-unprod [<=] 13.5	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]
   add-sqr-sqrt [<=] 26.8	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{-x}}{s}}}} \]
   add-sqr-sqrt [=>] 26.8	\[ \frac{1}{1 + \frac{1}{\color{blue}{\sqrt{e^{\frac{-x}{s}}} \cdot \sqrt{e^{\frac{-x}{s}}}}}} \]
   associate-/r* [=>] 26.8	\[ \frac{1}{1 + \color{blue}{\frac{\frac{1}{\sqrt{e^{\frac{-x}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}}} \]
   add-sqr-sqrt [=>] 13.5	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqrt-unprod [=>] 24.9	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqr-neg [=>] 24.9	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\sqrt{\color{blue}{x \cdot x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqrt-unprod [<=] 11.4	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   add-sqr-sqrt [<=] 22.7	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{x}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.7	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   add-exp-log [=>] 99.7	\[ \frac{1}{\color{blue}{e^{\log \left(1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}\right)}}} \]
   rec-exp [=>] 99.7	\[ \color{blue}{e^{-\log \left(1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}\right)}} \]
   log1p-def [=>] 99.7	\[ e^{-\color{blue}{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}\right)}} \]
   associate-/l/ [=>] 99.8	\[ e^{-\mathsf{log1p}\left(\color{blue}{\frac{1}{\sqrt{e^{\frac{x}{s}}} \cdot \sqrt{e^{\frac{x}{s}}}}}\right)} \]
   add-sqr-sqrt [<=] 99.8	\[ e^{-\mathsf{log1p}\left(\frac{1}{\color{blue}{e^{\frac{x}{s}}}}\right)} \]
   rec-exp [=>] 99.8	\[ e^{-\mathsf{log1p}\left(\color{blue}{e^{-\frac{x}{s}}}\right)} \]

4. Simplified 99.8%

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

   [Start] 99.8	\[ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \]
   distribute-neg-frac [=>] 99.8	\[ e^{-\mathsf{log1p}\left(e^{\color{blue}{\frac{-x}{s}}}\right)} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	3456

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

Alternative 2
Accuracy	63.2%
Cost	644

\[\begin{array}{l} \mathbf{if}\;-x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \frac{0.5}{s \cdot s}\right) + \left(2 - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 3
Accuracy	65.8%
Cost	420

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{-x}\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	49.3%
Cost	388

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

Alternative 5
Accuracy	47.7%
Cost	324

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

Alternative 6
Accuracy	46.4%
Cost	228

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Accuracy	46.4%
Cost	164

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

Alternative 8
Accuracy	35.2%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))