Logistic function

Average Error: 0.18% → 0.19%

Time: 14.8s

Precision: binary32

Cost: 3456

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

Math

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

↓

\[\frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)} \]

FPCore

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

↓

(FPCore (x s) :precision binary32 (/ 1.0 (+ 2.0 (expm1 (/ (- 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 / (2.0f + expm1f((-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(2.0) + expm1(Float32(Float32(-x) / s))))
end

Derivation

1. Initial program 0.18

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

2. Applied egg-rr 0.2

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

3. Applied egg-rr 0.75

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

4. Simplified 0.19

   \[\leadsto \color{blue}{\frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)}} \]
   Proof

   [Start] 0.75	\[ {\left(1 + e^{-\frac{x}{s}}\right)}^{-0.5} \cdot {\left(1 + e^{-\frac{x}{s}}\right)}^{-0.5} \]
   pow-sqr [=>] 0.18	\[ \color{blue}{{\left(1 + e^{-\frac{x}{s}}\right)}^{\left(2 \cdot -0.5\right)}} \]
   metadata-eval [=>] 0.18	\[ {\left(1 + e^{-\frac{x}{s}}\right)}^{\color{blue}{-1}} \]
   unpow-1 [=>] 0.18	\[ \color{blue}{\frac{1}{1 + e^{-\frac{x}{s}}}} \]
   +-commutative [=>] 0.18	\[ \frac{1}{\color{blue}{e^{-\frac{x}{s}} + 1}} \]
   metadata-eval [<=] 0.18	\[ \frac{1}{e^{-\frac{x}{s}} + \color{blue}{\left(2 - 1\right)}} \]
   associate--l+ [<=] 0.2	\[ \frac{1}{\color{blue}{\left(e^{-\frac{x}{s}} + 2\right) - 1}} \]
   +-commutative [<=] 0.2	\[ \frac{1}{\color{blue}{\left(2 + e^{-\frac{x}{s}}\right)} - 1} \]
   associate--l+ [=>] 0.18	\[ \frac{1}{\color{blue}{2 + \left(e^{-\frac{x}{s}} - 1\right)}} \]
   mul-1-neg [<=] 0.18	\[ \frac{1}{2 + \left(e^{\color{blue}{-1 \cdot \frac{x}{s}}} - 1\right)} \]
   expm1-def [=>] 0.19	\[ \frac{1}{2 + \color{blue}{\mathsf{expm1}\left(-1 \cdot \frac{x}{s}\right)}} \]
   mul-1-neg [=>] 0.19	\[ \frac{1}{2 + \mathsf{expm1}\left(\color{blue}{-\frac{x}{s}}\right)} \]
   distribute-neg-frac [=>] 0.19	\[ \frac{1}{2 + \mathsf{expm1}\left(\color{blue}{\frac{-x}{s}}\right)} \]

5. Final simplification 0.19

   \[\leadsto \frac{1}{2 + \mathsf{expm1}\left(\frac{-x}{s}\right)} \]

Alternatives

Alternative 1
Error	0.18%
Cost	3456

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

Alternative 2
Error	7.71%
Cost	808

\[\begin{array}{l} \mathbf{if}\;-x \leq -5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\left(1 + \frac{x}{s}\right) + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + x \cdot \frac{-1}{\frac{s}{\frac{-0.5}{\frac{s}{x}}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 3
Error	6.09%
Cost	808

\[\begin{array}{l} \mathbf{if}\;-x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\left(1 + \frac{x}{s}\right) + 0.5 \cdot \left(x \cdot \frac{\frac{x}{s}}{s}\right)}}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 4
Error	7.4%
Cost	744

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \mathbf{elif}\;x \leq -3.0000000095132306 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 5
Error	7.18%
Cost	680

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \mathbf{elif}\;x \leq -3.0000000095132306 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + x \cdot \left(x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 6
Error	8.17%
Cost	552

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -400:\\ \;\;\;\;1 + \frac{s \cdot s}{x \cdot x} \cdot -2\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 7
Error	7.58%
Cost	516

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 8
Error	34.81%
Cost	388

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 9
Error	51.7%
Cost	356

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

Alternative 10
Error	53.05%
Cost	164

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11
Error	64.53%
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))