Logistic function

Average Accuracy: 99.8% → 99.8%

Time: 15.6s

Precision: binary32

Cost: 16256

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

Math

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

↓

\[{\left(\mathsf{hypot}\left(1, {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s} \cdot 2\right)}\right)\right)}^{-2} \]

FPCore

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

↓

(FPCore (x s)
 :precision binary32
 (pow (hypot 1.0 (pow (sqrt (exp -0.5)) (* (/ x s) 2.0))) -2.0))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

↓

float code(float x, float s) {
	return powf(hypotf(1.0f, powf(sqrtf(expf(-0.5f)), ((x / s) * 2.0f))), -2.0f);
}

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 hypot(Float32(1.0), (sqrt(exp(Float32(-0.5))) ^ Float32(Float32(x / s) * Float32(2.0)))) ^ Float32(-2.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((-x / s)));
end

↓

function tmp = code(x, s)
	tmp = hypot(single(1.0), (sqrt(exp(single(-0.5))) ^ ((x / s) * single(2.0)))) ^ single(-2.0);
end

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}} \]
   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 [=>] 99.8	\[ \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
   add-sqr-sqrt [=>] 49.8	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}} \]
   sqrt-unprod [=>] 60.7	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{x \cdot x}}}{s}}}} \]
   sqr-neg [<=] 60.7	\[ \frac{1}{1 + \frac{1}{e^{\frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{s}}}} \]
   sqrt-unprod [<=] 13.2	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]
   add-sqr-sqrt [<=] 26.2	\[ \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{-x}}{s}}}} \]
   add-sqr-sqrt [=>] 26.2	\[ \frac{1}{1 + \frac{1}{\color{blue}{\sqrt{e^{\frac{-x}{s}}} \cdot \sqrt{e^{\frac{-x}{s}}}}}} \]
   associate-/r* [=>] 26.2	\[ \frac{1}{1 + \color{blue}{\frac{\frac{1}{\sqrt{e^{\frac{-x}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}}} \]
   add-sqr-sqrt [=>] 13.2	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqrt-unprod [=>] 24.3	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqr-neg [=>] 24.3	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\sqrt{\color{blue}{x \cdot x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   sqrt-unprod [<=] 11.1	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]
   add-sqr-sqrt [<=] 22.0	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{\color{blue}{x}}{s}}}}}{\sqrt{e^{\frac{-x}{s}}}}} \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-1}} \]
   Proof

   [Start] 99.8	\[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   inv-pow [=>] 99.8	\[ \color{blue}{{\left(1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}\right)}^{-1}} \]
   add-sqr-sqrt [=>] 99.2	\[ {\color{blue}{\left(\sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \cdot \sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}\right)}}^{-1} \]
   metadata-eval [<=] 99.2	\[ {\left(\sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \cdot \sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}\right)}^{\color{blue}{\left(-1\right)}} \]
   unpow-prod-down [=>] 99.1	\[ \color{blue}{{\left(\sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}\right)}^{\left(-1\right)} \cdot {\left(\sqrt{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}\right)}^{\left(-1\right)}} \]

4. Simplified 99.8%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}^{-2}} \]
   Proof

   [Start] 99.2	\[ {\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-1} \]
   pow-sqr [=>] 99.8	\[ \color{blue}{{\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{\left(2 \cdot -1\right)}} \]
   exp-prod [=>] 99.8	\[ {\left(\mathsf{hypot}\left(1, \color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}\right)\right)}^{\left(2 \cdot -1\right)} \]
   metadata-eval [=>] 99.8	\[ {\left(\mathsf{hypot}\left(1, {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}^{\color{blue}{-2}} \]

5. Applied egg-rr 99.7%

   \[\leadsto {\left(\mathsf{hypot}\left(1, \color{blue}{{\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)}}\right)\right)}^{-2} \]
   Proof

   [Start] 99.8	\[ {\left(\mathsf{hypot}\left(1, {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}^{-2} \]
   add-sqr-sqrt [=>] 99.8	\[ {\left(\mathsf{hypot}\left(1, {\color{blue}{\left(\sqrt{e^{-0.5}} \cdot \sqrt{e^{-0.5}}\right)}}^{\left(\frac{x}{s}\right)}\right)\right)}^{-2} \]
   unpow-prod-down [=>] 99.7	\[ {\left(\mathsf{hypot}\left(1, \color{blue}{{\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)}}\right)\right)}^{-2} \]

6. Simplified 99.8%

   \[\leadsto {\left(\mathsf{hypot}\left(1, \color{blue}{{\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s} \cdot 2\right)}}\right)\right)}^{-2} \]
   Proof

   [Start] 99.7	\[ {\left(\mathsf{hypot}\left(1, {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}^{-2} \]
   pow-sqr [=>] 99.8	\[ {\left(\mathsf{hypot}\left(1, \color{blue}{{\left(\sqrt{e^{-0.5}}\right)}^{\left(2 \cdot \frac{x}{s}\right)}}\right)\right)}^{-2} \]
   *-commutative [=>] 99.8	\[ {\left(\mathsf{hypot}\left(1, {\left(\sqrt{e^{-0.5}}\right)}^{\color{blue}{\left(\frac{x}{s} \cdot 2\right)}}\right)\right)}^{-2} \]

7. Final simplification 99.8%

   \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{x}{s} \cdot 2\right)}\right)\right)}^{-2} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	9824

\[{\left(\mathsf{hypot}\left(1, e^{-0.5 \cdot \frac{x}{s}}\right)\right)}^{-2} \]

Alternative 2
Accuracy	99.8%
Cost	3456

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

Alternative 3
Accuracy	63.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(x \cdot \frac{1}{s \cdot \frac{s \cdot 2}{x}} - \frac{x}{s}\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	64.0%
Cost	772

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

Alternative 5
Accuracy	63.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s} + \left(-2 + x \cdot \left(\frac{x}{s} \cdot \frac{-0.5}{s}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	62.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;-x \leq 1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 7
Accuracy	61.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;-x \leq 1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \frac{\frac{x \cdot x}{2}}{s \cdot s}\right)}\\ \end{array} \]

Alternative 8
Accuracy	59.7%
Cost	452

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

Alternative 9
Accuracy	61.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]

Alternative 10
Accuracy	61.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot 2\right)}{x \cdot x}\\ \end{array} \]

Alternative 11
Accuracy	49.3%
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 12
Accuracy	47.9%
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 13
Accuracy	46.0%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14
Accuracy	34.6%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))