Logistic distribution

Average Error: 0.1 → 0.1

Time: 17.1s

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	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	1.2
Cost	6688

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

Alternative 3
Error	1.2
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.0000000900125474 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s + \frac{s}{1 + \frac{x}{s}}}}{1 + t_0}\\ \end{array} \]

Alternative 4
Error	3.7
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\left(\frac{x}{s} + 2\right) + 0.5 \cdot \left(x \cdot \left(x \cdot \frac{-1}{s \cdot \left(-s\right)}\right)\right)}\\ \end{array} \]

Alternative 5
Error	1.5
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.4999999866463684 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t_0}\\ \end{array} \]

Alternative 6
Error	6.6
Cost	772

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\left(\frac{x}{s} + 2\right) + 0.5 \cdot \left(x \cdot \left(x \cdot \frac{-1}{s \cdot \left(-s\right)}\right)\right)}\\ \end{array} \]

Alternative 7
Error	6.3
Cost	676

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\left(\frac{x}{s} + 2\right) + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 8
Error	6.4
Cost	676

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\left(\frac{x}{s} + 2\right) + \frac{x \cdot 0.5}{\frac{s \cdot s}{x}}}\\ \end{array} \]

Alternative 9
Error	5.7
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999999279835 \cdot 10^{-23} \lor \neg \left(x \leq 4.999999999099794 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + s \cdot 4}\\ \end{array} \]

Alternative 10
Error	10.8
Cost	352

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

Alternative 11
Error	11.9
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1999999952050366 \cdot 10^{-11} \lor \neg \left(x \leq 3.999999886872274 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 12
Error	11.9
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1999999952050366 \cdot 10^{-11}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 13
Error	23.1
Cost	96

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

Reproduce

herbie shell --seed 2023053 
(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))))))