Logistic distribution

Average Error: 0.2 → 0.2

Time: 15.6s

Precision: binary32

Cost: 3552

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

Math

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

↓

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

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) (+ -2.0 (* -2.0 (cosh (/ 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) / (-2.0f + (-2.0f * coshf((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) / ((-2.0e0) + ((-2.0e0) * cosh((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(Float32(-1.0) / s) / Float32(Float32(-2.0) + Float32(Float32(-2.0) * cosh(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(-2.0) + (single(-2.0) * cosh((x / s))));
end

Derivation

1. Initial program 0.2

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

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] 0.2	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [<=] 0.2	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 0.2	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 0.2	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 0.2	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 0.2	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 11.7

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

4. Taylor expanded in s around 0 11.7

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

5. Simplified 0.1

   \[\leadsto \frac{1}{\frac{s}{e^{\frac{x}{s}}} + \color{blue}{s \cdot \left(e^{\frac{x}{s}} + 2\right)}} \]
   Proof

   [Start] 11.7	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + \left(1 + e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right)\right) \cdot s} \]
   *-commutative [=>] 11.7	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + \color{blue}{s \cdot \left(1 + e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right)\right)}} \]
   +-commutative [=>] 11.7	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \color{blue}{\left(e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right) + 1\right)}} \]
   distribute-lft-in [=>] 23.2	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(\color{blue}{\left(e^{\frac{x}{s}} \cdot 1 + e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}}\right)} + 1\right)} \]
   associate-+l+ [=>] 23.2	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \color{blue}{\left(e^{\frac{x}{s}} \cdot 1 + \left(e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}} + 1\right)\right)}} \]
   *-rgt-identity [=>] 23.2	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(\color{blue}{e^{\frac{x}{s}}} + \left(e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}} + 1\right)\right)} \]
   rgt-mult-inverse [=>] 0.1	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(e^{\frac{x}{s}} + \left(\color{blue}{1} + 1\right)\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(e^{\frac{x}{s}} + \color{blue}{2}\right)} \]

6. Taylor expanded in s around 0 0.1

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{x}{s}} + \left(2 + \frac{1}{e^{\frac{x}{s}}}\right)\right)}} \]

7. Simplified 0.2

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

   [Start] 0.1	\[ \frac{1}{s \cdot \left(e^{\frac{x}{s}} + \left(2 + \frac{1}{e^{\frac{x}{s}}}\right)\right)} \]
   associate-/r* [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{s}}{e^{\frac{x}{s}} + \left(2 + \frac{1}{e^{\frac{x}{s}}}\right)}} \]
   associate-+r+ [=>] 0.2	\[ \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + 2\right) + \frac{1}{e^{\frac{x}{s}}}}} \]
   +-commutative [=>] 0.2	\[ \frac{\frac{1}{s}}{\color{blue}{\left(2 + e^{\frac{x}{s}}\right)} + \frac{1}{e^{\frac{x}{s}}}} \]
   exp-neg [<=] 0.2	\[ \frac{\frac{1}{s}}{\left(2 + e^{\frac{x}{s}}\right) + \color{blue}{e^{-\frac{x}{s}}}} \]
   distribute-neg-frac [=>] 0.2	\[ \frac{\frac{1}{s}}{\left(2 + e^{\frac{x}{s}}\right) + e^{\color{blue}{\frac{-x}{s}}}} \]

8. Applied egg-rr 0.2

   \[\leadsto \color{blue}{-\frac{\frac{1}{s}}{-2 - 2 \cdot \cosh \left(\frac{x}{s}\right)}} \]

9. Final simplification 0.2

   \[\leadsto \frac{\frac{-1}{s}}{-2 + -2 \cdot \cosh \left(\frac{x}{s}\right)} \]

Alternatives

Alternative 1
Error	1.4
Cost	3588

\[\begin{array}{l} \mathbf{if}\;x \leq -5.00000011871114 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + e^{\frac{-x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + e^{\frac{x}{s}}}\\ \end{array} \]

Alternative 2
Error	3.0
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + t_0}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	3552

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

Alternative 4
Error	3.5
Cost	3364

\[\begin{array}{l} \mathbf{if}\;x \leq -1.199999960510567 \cdot 10^{-10}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{-4 - \frac{x}{\frac{s \cdot s}{x}}} \cdot \frac{-1}{s}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \end{array} \]

Alternative 5
Error	4.2
Cost	617

\[\begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873 \lor \neg \left(x \leq 0.009999999776482582\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-4 - \frac{x}{\frac{s \cdot s}{x}}} \cdot \frac{-1}{s}\\ \end{array} \]

Alternative 6
Error	5.8
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873 \lor \neg \left(x \leq 0.009999999776482582\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 7
Error	5.4
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873 \lor \neg \left(x \leq 0.009999999776482582\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 8
Error	4.1
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873 \lor \neg \left(x \leq 0.009999999776482582\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{x \cdot \frac{x}{s \cdot s} + 4}\\ \end{array} \]

Alternative 9
Error	6.5
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-13}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 10
Error	12.2
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 11
Error	12.2
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;s \cdot \frac{-1}{-x \cdot x}\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 12
Error	12.2
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9} \lor \neg \left(x \leq 0.009999999776482582\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Error	12.2
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 14
Error	23.3
Cost	96

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

Reproduce

herbie shell --seed 2023060 
(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))))))