Logistic distribution

Average Error: 0.41% → 0.41%

Time: 15.0s

Precision: binary32

Cost: 6944

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\frac{s + \frac{s}{t_0}}{\frac{1}{1 + t_0}}} \end{array} \]

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
 (let* ((t_0 (exp (/ x s)))) (/ 1.0 (/ (+ s (/ s t_0)) (/ 1.0 (+ 1.0 t_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) {
	float t_0 = expf((x / s));
	return 1.0f / ((s + (s / t_0)) / (1.0f / (1.0f + t_0)));
}

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
    real(4) :: t_0
    t_0 = exp((x / s))
    code = 1.0e0 / ((s + (s / t_0)) / (1.0e0 / (1.0e0 + t_0)))
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)
	t_0 = exp(Float32(x / s))
	return Float32(Float32(1.0) / Float32(Float32(s + Float32(s / t_0)) / Float32(Float32(1.0) / Float32(Float32(1.0) + t_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)
	t_0 = exp((x / s));
	tmp = single(1.0) / ((s + (s / t_0)) / (single(1.0) / (single(1.0) + t_0)));
end

Derivation

1. Initial program 0.41

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

2. Simplified 0.42

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

   [Start] 0.41	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [<=] 0.39	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 0.39	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 0.39	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 0.39	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 0.39	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l* [=>] 0.46	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 0.44	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Applied egg-rr 0.41

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

4. Final simplification 0.41

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

Alternatives

Alternative 1
Error	0.38%
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	3.54%
Cost	6688

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

Alternative 3
Error	5.09%
Cost	6656

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

Alternative 4
Error	13.92%
Cost	3624

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 5
Error	13.92%
Cost	3624

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, s, \frac{x}{\frac{s}{x}}\right)}\\ \mathbf{elif}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 6
Error	13.92%
Cost	3624

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\\ \mathbf{elif}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 7
Error	4.2%
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{s + \frac{s}{t_0}}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 8
Error	21.15%
Cost	813

\[\begin{array}{l} \mathbf{if}\;x \leq -10000 \lor \neg \left(x \leq -5.000000156871975 \cdot 10^{-23}\right) \land x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9
Error	34.46%
Cost	352

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

Alternative 10
Error	37.15%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513 \lor \neg \left(x \leq 0.00019999999494757503\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11
Error	37.19%
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 12
Error	72.57%
Cost	96

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

Reproduce

herbie shell --seed 2023102 
(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))))))