Logistic distribution

Average Error: 0.1 → 0.1

Time: 14.8s

Precision: binary32

Cost: 19808

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

Math

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

↓

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

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 (/ (- (fabs x)) s))))
   (/ t_0 (* (+ s (/ s (exp (/ (fabs x) s)))) (+ t_0 1.0)))))

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((-fabsf(x) / s));
	return t_0 / ((s + (s / expf((fabsf(x) / s)))) * (t_0 + 1.0f));
}

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
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / ((s + (s / exp((abs(x) / s)))) * (t_0 + 1.0e0))
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)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(s + Float32(s / exp(Float32(abs(x) / s)))) * Float32(t_0 + Float32(1.0))))
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((-abs(x) / s));
	tmp = t_0 / ((s + (s / exp((abs(x) / s)))) * (t_0 + single(1.0)));
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 0.1

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	6944

\[\frac{1}{\frac{s + \frac{s}{e^{\frac{x}{s}}}}{\frac{1}{2 + \mathsf{expm1}\left(\frac{x}{s}\right)}}} \]

Alternative 2
Error	0.1
Cost	6880

\[\frac{1}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(2 + \mathsf{expm1}\left(\frac{x}{s}\right)\right)} \]

Alternative 3
Error	0.1
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 4
Error	0.2
Cost	6848

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

Alternative 5
Error	1.2
Cost	6688

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

Alternative 6
Error	1.6
Cost	6656

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

Alternative 7
Error	1.2
Cost	3876

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{s}{t_0} + s \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + t_0}{\frac{1}{s + \frac{s}{1 + \frac{x}{s}}}}}\\ \end{array} \]

Alternative 8
Error	1.2
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{s}{t_0} + s \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 9
Error	3.7
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)}\\ \end{array} \]

Alternative 10
Error	5.6
Cost	552

\[\begin{array}{l} \mathbf{if}\;x \leq -2.999999970665357 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 11
Error	5.8
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -2.999999970665357 \cdot 10^{-10} \lor \neg \left(x \leq 1.0000000036274937 \cdot 10^{-15}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12
Error	6.1
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15} \lor \neg \left(x \leq 1.0000000036274937 \cdot 10^{-15}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Error	11.5
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15} \lor \neg \left(x \leq 5.000000058430487 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Error	11.9
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 15
Error	11.9
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15} \lor \neg \left(x \leq 5.000000058430487 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Error	11.9
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9999998100067255 \cdot 10^{-15}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 17
Error	23.2
Cost	96

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

Reproduce

herbie shell --seed 2023041 
(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))))))