Logistic distribution

Average Error: 0.2 → 0.2

Time: 16.0s

Precision: binary32

Cost: 13248

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

Math

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

↓

\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

FPCore

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

↓

(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (+ (exp (/ (fabs x) s)) (+ (exp (/ (fabs x) (- s))) 2.0))))

C

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

↓

float code(float x, float s) {
	return (1.0f / s) / (expf((fabsf(x) / s)) + (expf((fabsf(x) / -s)) + 2.0f));
}

Fortran

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

↓

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (exp((abs(x) / s)) + (exp((abs(x) / -s)) + 2.0e0))
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(s * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))) * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))))
end

↓

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / s)) + Float32(exp(Float32(abs(x) / Float32(-s))) + Float32(2.0))))
end

MATLAB

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

↓

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((abs(x) / s)) + (exp((abs(x) / -s)) + single(2.0)));
end

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 0.2	\[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*l* [=>] 0.2	\[ \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   times-frac [=>] 0.2	\[ \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]
   distribute-frac-neg [=>] 0.2	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]
   exp-neg [=>] 0.2	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]
   associate-/r/ [=>] 0.2	\[ \frac{\frac{1}{s}}{\color{blue}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]
   /-rgt-identity [=>] 0.2	\[ \frac{\frac{1}{s}}{\color{blue}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{1}{s}}{\color{blue}{e^{\frac{\left|x\right|}{s}} \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   distribute-rgt-in [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\left(1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right) + e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   *-lft-identity [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot \left(\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)} + e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   associate-+l+ [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\left(1 + \left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)}} \]
   distribute-lft-in [=>] 23.4	\[ \frac{\frac{1}{s}}{\color{blue}{e^{\frac{\left|x\right|}{s}} \cdot 1 + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   *-rgt-identity [=>] 23.4	\[ \frac{\frac{1}{s}}{\color{blue}{e^{\frac{\left|x\right|}{s}}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   distribute-lft-in [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)}} \]
   distribute-frac-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} \cdot e^{\color{blue}{-\frac{\left|x\right|}{s}}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   exp-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   rgt-mult-inverse [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{1} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   *-commutative [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{-\left|x\right|}{s}}\right)}\right)} \]
   +-commutative [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + e^{\frac{\left|x\right|}{s}} \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   distribute-rgt1-in [<=] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)}\right)} \]
   distribute-lft-in [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + \color{blue}{\left(e^{\frac{\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)}\right)} \]
   distribute-frac-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + \left(e^{\frac{\left|x\right|}{s}} \cdot e^{\color{blue}{-\frac{\left|x\right|}{s}}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   exp-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + \left(e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   rgt-mult-inverse [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(1 + \left(\color{blue}{1} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)\right)} \]
   associate-+r+ [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(\left(1 + 1\right) + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   metadata-eval [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{2} + e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   +-commutative [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right) + 2\right)}} \]
   prod-exp [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} \cdot \color{blue}{e^{\frac{-\left|x\right|}{s} + \frac{-\left|x\right|}{s}}} + 2\right)} \]
   prod-exp [=>] 8.3	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{e^{\frac{\left|x\right|}{s} + \left(\frac{-\left|x\right|}{s} + \frac{-\left|x\right|}{s}\right)}} + 2\right)} \]
   +-commutative [<=] 8.3	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{\left(\frac{-\left|x\right|}{s} + \frac{-\left|x\right|}{s}\right) + \frac{\left|x\right|}{s}}} + 2\right)} \]
   prod-exp [<=] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{e^{\frac{-\left|x\right|}{s} + \frac{-\left|x\right|}{s}} \cdot e^{\frac{\left|x\right|}{s}}} + 2\right)} \]
   prod-exp [<=] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} \cdot e^{\frac{-\left|x\right|}{s}}\right)} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   prod-exp [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{e^{\frac{-\left|x\right|}{s} + \frac{-\left|x\right|}{s}}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   distribute-frac-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{\left(-\frac{\left|x\right|}{s}\right)} + \frac{-\left|x\right|}{s}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   neg-mul-1 [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}} + \frac{-\left|x\right|}{s}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   distribute-frac-neg [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{-1 \cdot \frac{\left|x\right|}{s} + \color{blue}{\left(-\frac{\left|x\right|}{s}\right)}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   neg-mul-1 [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{-1 \cdot \frac{\left|x\right|}{s} + \color{blue}{-1 \cdot \frac{\left|x\right|}{s}}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   distribute-rgt-out [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{\frac{\left|x\right|}{s} \cdot \left(-1 + -1\right)}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   exp-prod [=>] 23.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{{\left(e^{\frac{\left|x\right|}{s}}\right)}^{\left(-1 + -1\right)}} \cdot e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   pow-plus [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{{\left(e^{\frac{\left|x\right|}{s}}\right)}^{\left(\left(-1 + -1\right) + 1\right)}} + 2\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left({\left(e^{\frac{\left|x\right|}{s}}\right)}^{\left(\color{blue}{-2} + 1\right)} + 2\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left({\left(e^{\frac{\left|x\right|}{s}}\right)}^{\color{blue}{-1}} + 2\right)} \]
   unpow-1 [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + 2\right)} \]
   exp-neg [<=] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{e^{-\frac{\left|x\right|}{s}}} + 2\right)} \]
   neg-mul-1 [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}} + 2\right)} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{\frac{\left|x\right|}{s} \cdot -1}} + 2\right)} \]
   associate-/r/ [<=] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\color{blue}{\frac{\left|x\right|}{\frac{s}{-1}}}} + 2\right)} \]
   *-lft-identity [<=] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{\frac{\color{blue}{1 \cdot s}}{-1}}} + 2\right)} \]
   associate-/l* [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{\color{blue}{\frac{1}{\frac{-1}{s}}}}} + 2\right)} \]
   associate-/r/ [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{\color{blue}{\frac{1}{-1} \cdot s}}} + 2\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{\color{blue}{-1} \cdot s}} + 2\right)} \]
   mul-1-neg [=>] 0.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 2\right)} \]

3. Final simplification 0.2

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

Alternatives

Alternative 1
Error	0.2
Cost	6880

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)} \end{array} \]

Alternative 2
Error	1.3
Cost	6688

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

Alternative 3
Error	1.3
Cost	6688

\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]

Alternative 4
Error	1.8
Cost	6656

\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]

Alternative 5
Error	1.2
Cost	4132

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s + \frac{s}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{\left(1 + \frac{x}{s}\right) + 0.5 \cdot \frac{x}{\frac{s}{\frac{x}{s}}}}\right)}\\ \end{array} \]

Alternative 6
Error	1.4
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s + \frac{s}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{1 + \frac{x}{s}}\right)}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	3620

\[\begin{array}{l} \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1 + \mathsf{expm1}\left(\frac{x}{s}\right)}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s\right)}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s + \frac{s}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + s\right)}\\ \end{array} \]

Alternative 9
Error	1.7
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s + \frac{s}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot t_0}\\ \end{array} \]

Alternative 10
Error	1.8
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;\frac{1 + \mathsf{expm1}\left(\frac{x}{s}\right)}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}\\ \end{array} \]

Alternative 11
Error	4.4
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 12
Error	1.8
Cost	3492

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot t_0}\\ \end{array} \]

Alternative 13
Error	1.8
Cost	3492

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.799999975007332 \cdot 10^{-35}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]

Alternative 14
Error	5.2
Cost	1001

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 2.0000000233721948 \cdot 10^{-7}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{1 + \frac{x}{s}}\right) \cdot \left(2 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 15
Error	4.9
Cost	1001

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 2.0000000233721948 \cdot 10^{-7}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{1 + \frac{x}{s}}\right) \cdot \left(\frac{x}{s} + \left(2 + 0.5 \cdot \frac{x}{\frac{s}{\frac{x}{s}}}\right)\right)}\\ \end{array} \]

Alternative 16
Error	5.9
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999858590343 \cdot 10^{-10} \lor \neg \left(x \leq 3.007000030919027 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \]

Alternative 17
Error	6.2
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(x \leq 1.99999996490334 \cdot 10^{-13}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 18
Error	11.7
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7} \lor \neg \left(x \leq 0.00019999999494757503\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 19
Error	23.5
Cost	96

\[\frac{0.25}{s} \]

Reproduce

herbie shell --seed 2022356 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))