Logistic function

Average Error: 0.1 → 0.1

Time: 14.6s

Precision: binary32

Cost: 10048

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (/ (pow (exp -0.5) (/ x s)) (exp (* (/ x s) 0.5))))))

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 + (powf(expf(-0.5f), (x / s)) / expf(((x / s) * 0.5f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((-x / s)))
end function

↓

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + ((exp((-0.5e0)) ** (x / s)) / exp(((x / s) * 0.5e0))))
end function

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((exp(Float32(-0.5)) ^ Float32(x / s)) / exp(Float32(Float32(x / s) * Float32(0.5))))))
end

MATLAB

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

↓

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Taylor expanded in x around inf 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{1}{1 + \frac{\sqrt{\frac{1}{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   unpow-1 [<=] 0.1	\[ \frac{1}{1 + \frac{\sqrt{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-1}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   metadata-eval [<=] 0.1	\[ \frac{1}{1 + \frac{\sqrt{{\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   pow-sqr [<=] 0.1	\[ \frac{1}{1 + \frac{\sqrt{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.5} \cdot {\left(e^{\frac{x}{s}}\right)}^{-0.5}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   rem-sqrt-square [=>] 0.1	\[ \frac{1}{1 + \frac{\color{blue}{\left|{\left(e^{\frac{x}{s}}\right)}^{-0.5}\right|}}{\sqrt{e^{\frac{x}{s}}}}} \]
   sqr-pow [=>] 0.1	\[ \frac{1}{1 + \frac{\left|\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(e^{\frac{x}{s}}\right)}^{\left(\frac{-0.5}{2}\right)}}\right|}{\sqrt{e^{\frac{x}{s}}}}} \]
   fabs-sqr [=>] 0.1	\[ \frac{1}{1 + \frac{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(e^{\frac{x}{s}}\right)}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   sqr-pow [<=] 0.1	\[ \frac{1}{1 + \frac{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.5}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   exp-prod [<=] 0.1	\[ \frac{1}{1 + \frac{\color{blue}{e^{\frac{x}{s} \cdot -0.5}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   *-commutative [<=] 0.1	\[ \frac{1}{1 + \frac{e^{\color{blue}{-0.5 \cdot \frac{x}{s}}}}{\sqrt{e^{\frac{x}{s}}}}} \]
   exp-prod [=>] 0.1	\[ \frac{1}{1 + \frac{\color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}}{\sqrt{e^{\frac{x}{s}}}}} \]

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.0
Cost	9760

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

Alternative 2
Error	0.1
Cost	3520

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

Alternative 3
Error	0.1
Cost	3456

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

Alternative 4
Error	1.5
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 5
Error	10.0
Cost	196

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

Alternative 6
Error	20.9
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))