Logistic distribution

Average Error: 0.44% → 0.44%

Time: 17.3s

Precision: binary32

Cost: 6912

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

Math

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

↓

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

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 (* (+ 1.0 (exp (/ (- x) s))) (+ 1.0 (exp (/ x s)))))))

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 * ((1.0f + expf((-x / s))) * (1.0f + expf((x / s)))));
}

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 * ((1.0e0 + exp((-x / s))) * (1.0e0 + exp((x / s)))))
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(1.0) / Float32(s * Float32(Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) * Float32(Float32(1.0) + exp(Float32(x / s))))))
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 * ((single(1.0) + exp((-x / s))) * (single(1.0) + exp((x / s)))));
end

Derivation

1. Initial program 0.44

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

2. Simplified 0.44

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

   [Start] 0.44	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [<=] 0.42	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 0.42	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 0.42	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 0.42	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 0.42	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l* [=>] 0.48	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 0.48	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Applied egg-rr 37.06

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

4. Simplified 0.42

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

   [Start] 37.06	\[ \frac{1}{s + \left(\frac{s}{e^{\frac{x}{s}}} \cdot \left(e^{\frac{x}{s}} + 1\right) + s \cdot e^{\frac{x}{s}}\right)} \]
   +-commutative [=>] 37.06	\[ \frac{1}{s + \color{blue}{\left(s \cdot e^{\frac{x}{s}} + \frac{s}{e^{\frac{x}{s}}} \cdot \left(e^{\frac{x}{s}} + 1\right)\right)}} \]
   distribute-rgt-in [=>] 72.73	\[ \frac{1}{s + \left(s \cdot e^{\frac{x}{s}} + \color{blue}{\left(e^{\frac{x}{s}} \cdot \frac{s}{e^{\frac{x}{s}}} + 1 \cdot \frac{s}{e^{\frac{x}{s}}}\right)}\right)} \]
   *-lft-identity [=>] 72.73	\[ \frac{1}{s + \left(s \cdot e^{\frac{x}{s}} + \left(e^{\frac{x}{s}} \cdot \frac{s}{e^{\frac{x}{s}}} + \color{blue}{\frac{s}{e^{\frac{x}{s}}}}\right)\right)} \]
   associate-+r+ [=>] 72.73	\[ \frac{1}{s + \color{blue}{\left(\left(s \cdot e^{\frac{x}{s}} + e^{\frac{x}{s}} \cdot \frac{s}{e^{\frac{x}{s}}}\right) + \frac{s}{e^{\frac{x}{s}}}\right)}} \]
   *-commutative [<=] 72.73	\[ \frac{1}{s + \left(\left(\color{blue}{e^{\frac{x}{s}} \cdot s} + e^{\frac{x}{s}} \cdot \frac{s}{e^{\frac{x}{s}}}\right) + \frac{s}{e^{\frac{x}{s}}}\right)} \]
   distribute-lft-in [<=] 36.05	\[ \frac{1}{s + \left(\color{blue}{e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)} + \frac{s}{e^{\frac{x}{s}}}\right)} \]
   *-rgt-identity [<=] 36.05	\[ \frac{1}{s + \left(\color{blue}{\left(e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right) \cdot 1} + \frac{s}{e^{\frac{x}{s}}}\right)} \]
   +-commutative [<=] 36.05	\[ \frac{1}{s + \color{blue}{\left(\frac{s}{e^{\frac{x}{s}}} + \left(e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right) \cdot 1\right)}} \]

5. Taylor expanded in s around 0 0.44

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

6. Applied egg-rr 0.44

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

7. Final simplification 0.44

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

Alternatives

Alternative 1
Error	0.42%
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 2
Error	3.9%
Cost	6688

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

Alternative 3
Error	5.15%
Cost	6656

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

Alternative 4
Error	3.9%
Cost	3684

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4 + \frac{x}{s} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 5
Error	4.32%
Cost	3652

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{1 + e^{\frac{-x}{s}}}{\frac{0.5}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 6
Error	4.41%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 7
Error	12.99%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(4 + \frac{1}{\frac{s}{x}}\right) + 0.5 \cdot \frac{1}{\frac{s \cdot s}{x \cdot x}}}\\ \end{array} \]

Alternative 8
Error	19.28%
Cost	3364

\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(4 + \frac{1}{\frac{s}{x}}\right) + 0.5 \cdot \frac{1}{\frac{s \cdot s}{x \cdot x}}}\\ \end{array} \]

Alternative 9
Error	29.34%
Cost	804

\[\begin{array}{l} \mathbf{if}\;x \leq -499999993495552:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(4 + \frac{1}{\frac{s}{x}}\right) + 0.5 \cdot \frac{1}{\frac{s \cdot s}{x \cdot x}}}\\ \end{array} \]

Alternative 10
Error	47.65%
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 9.999999747378752 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11
Error	48.5%
Cost	352

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

Alternative 12
Error	48.63%
Cost	288

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

Alternative 13
Error	72.71%
Cost	96

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

Alternative 14
Error	91.71%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))))))