Logistic distribution

Average Error: 0.1 → 0.1

Time: 20.0s

Precision: binary32

Cost: 13312

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

Math

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

↓

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

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 (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (fabs 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((-fabsf(x) / s)))) * (1.0f + expf((fabsf(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((-abs(x) / s)))) * (1.0e0 + exp((abs(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(Float32(s * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))) * Float32(Float32(1.0) + exp(Float32(abs(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((-abs(x) / s)))) * (single(1.0) + exp((abs(x) / s))));
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 0.1

   \[\leadsto \color{blue}{\frac{1}{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   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* [=>] 0.1	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 0.1	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Taylor expanded in s around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{1}{\left(s \cdot \left(1 + \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]
   exp-neg [<=] 0.1	\[ \frac{1}{\left(s \cdot \left(1 + \color{blue}{e^{-\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]
   distribute-neg-frac [=>] 0.1	\[ \frac{1}{\left(s \cdot \left(1 + e^{\color{blue}{\frac{-\left|x\right|}{s}}}\right)\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	0.7
Cost	13216

\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{s \cdot \left(e^{t_0} + \left|3 - t_0\right|\right)} \end{array} \]

Alternative 3
Error	1.2
Cost	13120

\[\frac{\frac{1}{s}}{{\left(\sqrt{e^{\frac{\left|x\right|}{s}} + 3}\right)}^{2}} \]

Alternative 4
Error	3.6
Cost	6952

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \mathbf{elif}\;\left|x\right| \leq 0.019999999552965164:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \end{array} \]

Alternative 5
Error	3.6
Cost	6952

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \mathbf{elif}\;\left|x\right| \leq 0.019999999552965164:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \end{array} \]

Alternative 6
Error	3.5
Cost	6729

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-22} \lor \neg \left(x \leq 1.999999936531045 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]

Alternative 7
Error	1.2
Cost	6720

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

Alternative 8
Error	1.2
Cost	6688

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

Alternative 9
Error	3.9
Cost	6665

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-22} \lor \neg \left(x \leq 1.999999936531045 \cdot 10^{-19}\right):\\ \;\;\;\;e^{\frac{-\left|x\right|}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \]

Alternative 10
Error	1.7
Cost	6656

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

Alternative 11
Error	6.2
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -4.99999991225835 \cdot 10^{-14} \lor \neg \left(x \leq 5.0000000843119176 \cdot 10^{-17}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12
Error	6.1
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -4.99999991225835 \cdot 10^{-14} \lor \neg \left(x \leq 5.000000018137469 \cdot 10^{-16}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \end{array} \]

Alternative 13
Error	6.6
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -4.99999991225835 \cdot 10^{-14} \lor \neg \left(x \leq 1.999999936531045 \cdot 10^{-19}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Error	11.7
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15
Error	11.3
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Error	11.3
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 17
Error	11.7
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 18
Error	23.1
Cost	96

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

Reproduce

herbie shell --seed 2023187 
(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))))))