Logistic function

Average Accuracy: 99.8% → 99.8%

Time: 13.5s

Precision: binary32

Cost: 16512

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

Math

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

↓

\[\frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{\frac{x}{s}}{\frac{s}{x}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]

FPCore

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

↓

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+ 1.0 (/ 1.0 (pow (pow E (cbrt (/ (/ x s) (/ s x)))) (cbrt (/ 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 / (1.0f + (1.0f / powf(powf(((float) M_E), cbrtf(((x / s) / (s / x)))), cbrtf((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(1.0) + Float32(Float32(1.0) / ((Float32(exp(1)) ^ cbrt(Float32(Float32(x / s) / Float32(s / x)))) ^ cbrt(Float32(x / s))))))
end

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

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

3. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{-2}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}}} \]
   Proof

   [Start] 99.8	\[ \frac{1}{1 + \frac{1}{e^{\frac{x}{s}}}} \]
   *-un-lft-identity [=>] 99.8	\[ \frac{1}{1 + \frac{1}{e^{\color{blue}{1 \cdot \frac{x}{s}}}}} \]
   exp-prod [=>] 99.8	\[ \frac{1}{1 + \frac{1}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}}} \]
   add-cube-cbrt [=>] 99.7	\[ \frac{1}{1 + \frac{1}{{\left(e^{1}\right)}^{\color{blue}{\left(\left(\sqrt[3]{\frac{x}{s}} \cdot \sqrt[3]{\frac{x}{s}}\right) \cdot \sqrt[3]{\frac{x}{s}}\right)}}}} \]
   pow-unpow [=>] 99.7	\[ \frac{1}{1 + \frac{1}{\color{blue}{{\left({\left(e^{1}\right)}^{\left(\sqrt[3]{\frac{x}{s}} \cdot \sqrt[3]{\frac{x}{s}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}}} \]
   exp-1-e [=>] 99.7	\[ \frac{1}{1 + \frac{1}{{\left({\color{blue}{e}}^{\left(\sqrt[3]{\frac{x}{s}} \cdot \sqrt[3]{\frac{x}{s}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   cbrt-unprod [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\color{blue}{\left(\sqrt[3]{\frac{x}{s} \cdot \frac{x}{s}}\right)}}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   clear-num [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{s}{x}}} \cdot \frac{x}{s}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   inv-pow [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\color{blue}{{\left(\frac{s}{x}\right)}^{-1}} \cdot \frac{x}{s}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   clear-num [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{s}{x}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   inv-pow [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{-1} \cdot \color{blue}{{\left(\frac{s}{x}\right)}^{-1}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   pow-prod-up [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\color{blue}{{\left(\frac{s}{x}\right)}^{\left(-1 + -1\right)}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   metadata-eval [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{\color{blue}{-2}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\color{blue}{\frac{\frac{x}{s}}{\frac{s}{x}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   Proof

   [Start] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{-2}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   metadata-eval [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{\color{blue}{\left(-1 - 1\right)}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   metadata-eval [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{{\left(\frac{s}{x}\right)}^{\left(\color{blue}{\frac{-2}{2}} - 1\right)}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   pow-div [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\color{blue}{\frac{{\left(\frac{s}{x}\right)}^{\left(\frac{-2}{2}\right)}}{{\left(\frac{s}{x}\right)}^{1}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   metadata-eval [=>] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{{\left(\frac{s}{x}\right)}^{\color{blue}{-1}}}{{\left(\frac{s}{x}\right)}^{1}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   inv-pow [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{\color{blue}{\frac{1}{\frac{s}{x}}}}{{\left(\frac{s}{x}\right)}^{1}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   clear-num [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{\color{blue}{\frac{x}{s}}}{{\left(\frac{s}{x}\right)}^{1}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]
   pow1 [<=] 99.8	\[ \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{\frac{x}{s}}{\color{blue}{\frac{s}{x}}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]

5. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + \frac{1}{{\left({e}^{\left(\sqrt[3]{\frac{\frac{x}{s}}{\frac{s}{x}}}\right)}\right)}^{\left(\sqrt[3]{\frac{x}{s}}\right)}}} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13216

\[\frac{1}{1 + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{-0.3333333333333333}\right)}^{\left(\frac{x}{s}\right)}} \]

Alternative 2
Accuracy	99.8%
Cost	3456

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

Alternative 3
Accuracy	97.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\left(1 + \frac{x}{s}\right) + \frac{\frac{x}{\frac{s}{x}} \cdot 0.5}{s}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Accuracy	97.4%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	97.4%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Accuracy	95.7%
Cost	516

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	95.6%
Cost	360

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	58.1%
Cost	100

\[\begin{array}{l} \mathbf{if}\;x \leq 5.000000097707407 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	35.8%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))