Logistic distribution

Average Error: 0.1 → 0.1

Time: 15.7s

Precision: binary32

Cost: 6880

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

Math

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

↓

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

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 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 + 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 + 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) + 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) + t_0));
end

Derivation

1. Initial program 0.1

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

2. Simplified 0.2

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [<=] 0.1	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 0.1	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 0.1	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 0.1	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l* [=>] 0.2	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 0.2	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Applied egg-rr 11.6

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

4. Simplified 0.1

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

   [Start] 11.6	\[ \frac{1}{s + \left(\frac{s}{e^{\frac{x}{s}}} + e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)} \]
   associate-+r+ [=>] 11.6	\[ \frac{1}{\color{blue}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) + e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}} \]
   *-lft-identity [<=] 11.6	\[ \frac{1}{\color{blue}{1 \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)} + e^{\frac{x}{s}} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)} \]
   distribute-rgt-in [<=] 0.1	\[ \frac{1}{\color{blue}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(1 + e^{\frac{x}{s}}\right)}} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.2
Cost	6688

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

Alternative 2
Error	1.8
Cost	6656

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

Alternative 3
Error	1.3
Cost	3684

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + \left(\frac{1}{t_0} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]

Alternative 4
Error	1.3
Cost	3652

\[\begin{array}{l} \mathbf{if}\;x \leq 1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + \left(2 + e^{\frac{-x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 5
Error	2.6
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]

Alternative 6
Error	3.0
Cost	3492

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]

Alternative 7
Error	5.1
Cost	3428

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4999999800084155 \cdot 10^{-24}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s}\\ \mathbf{elif}\;x \leq 6.000000240508405 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\left(s \cdot 4 + x \cdot \frac{0.5}{\frac{s}{x}}\right) + x \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 8
Error	6.9
Cost	813

\[\begin{array}{l} \mathbf{if}\;x \leq -20000000:\\ \;\;\;\;\frac{1}{x \cdot 0 + \left(s \cdot 4 + \frac{x \cdot \frac{x}{s}}{2}\right)}\\ \mathbf{elif}\;x \leq -1.9999999996399175 \cdot 10^{-23} \lor \neg \left(x \leq 6.000000240508405 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{0.25}{s}}{1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot 4 + x \cdot \frac{0.5}{\frac{s}{x}}\right) + x \cdot 0}\\ \end{array} \]

Alternative 9
Error	11.4
Cost	544

\[\frac{1}{x \cdot 0 + \left(s \cdot 4 + 0.5 \cdot \frac{x}{\frac{s}{x}}\right)} \]

Alternative 10
Error	11.4
Cost	544

\[\frac{1}{x \cdot 0 + \left(s \cdot 4 + \frac{x \cdot \frac{x}{s}}{2}\right)} \]

Alternative 11
Error	11.4
Cost	416

\[\frac{-1}{s \cdot -4 + \frac{x \cdot x}{s} \cdot -0.5} \]

Alternative 12
Error	12.1
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6} \lor \neg \left(x \leq 0.004999999888241291\right):\\ \;\;\;\;2 \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Error	23.4
Cost	96

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

Reproduce

herbie shell --seed 2023057 
(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))))))