Logistic distribution

Average Error: 0.2 → 0.2

Time: 15.4s

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

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	10080

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

Alternative 2
Error	0.2
Cost	6944

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

Alternative 3
Error	0.2
Cost	6880

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

Alternative 4
Error	1.1
Cost	6688

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

Alternative 5
Error	1.2
Cost	6688

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

Alternative 6
Error	1.1
Cost	4132

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -6.500000188762793 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{\frac{x}{s} + \left(1 + 0.5 \cdot \frac{\frac{x}{\frac{s}{x}}}{s}\right)}\right)}\\ \end{array} \]

Alternative 7
Error	3.8
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 8
Error	1.3
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -6.500000188762793 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 9
Error	6.0
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-21} \lor \neg \left(x \leq 1.9999999996399175 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \end{array} \]

Alternative 10
Error	11.0
Cost	352

\[\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4} \]

Alternative 11
Error	15.8
Cost	288

\[\frac{1}{s \cdot \left(\frac{x}{s} + 4\right)} \]

Alternative 12
Error	23.7
Cost	96

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

Reproduce

herbie shell --seed 2023027 
(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))))))