Logistic distribution

Average Error: 0.1 → 0.1

Time: 16.5s

Precision: binary32

Cost: 6880

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{s \cdot \left(t_0 + \left(2 + \frac{1}{t_0}\right)\right)} \end{array} \]

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 (+ t_0 (+ 2.0 (/ 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 * (t_0 + (2.0f + (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 * (t_0 + (2.0e0 + (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(s * Float32(t_0 + Float32(Float32(2.0) + 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 * (t_0 + (single(2.0) + (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 0.1

   \[\leadsto \color{blue}{\frac{1}{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   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. Applied egg-rr 11.6

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

4. Taylor expanded in s around 0 11.6

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

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] 11.6	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + \left(1 + e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right)\right) \cdot s} \]
   *-commutative [=>] 11.6	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + \color{blue}{s \cdot \left(1 + e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right)\right)}} \]
   +-commutative [=>] 11.6	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \color{blue}{\left(e^{\frac{x}{s}} \cdot \left(1 + \frac{1}{e^{\frac{x}{s}}}\right) + 1\right)}} \]
   distribute-lft-in [=>] 23.4	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(\color{blue}{\left(e^{\frac{x}{s}} \cdot 1 + e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}}\right)} + 1\right)} \]
   associate-+l+ [=>] 23.4	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \color{blue}{\left(e^{\frac{x}{s}} \cdot 1 + \left(e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}} + 1\right)\right)}} \]
   *-rgt-identity [=>] 23.4	\[ \frac{1}{\frac{s}{e^{\frac{x}{s}}} + s \cdot \left(\color{blue}{e^{\frac{x}{s}}} + \left(e^{\frac{x}{s}} \cdot \frac{1}{e^{\frac{x}{s}}} + 1\right)\right)} \]
   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. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	6848

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

Alternative 2
Error	0.2
Cost	6848

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

Alternative 3
Error	1.1
Cost	6688

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

Alternative 4
Error	1.2
Cost	6688

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

Alternative 5
Error	1.6
Cost	6656

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

Alternative 6
Error	0.9
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{\frac{s}{t_0} + s \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + \left(2 + \frac{1}{1 + \frac{x}{s}}\right)\right)}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	3620

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

Alternative 8
Error	1.1
Cost	3620

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

Alternative 9
Error	4.2
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5999999819625636 \cdot 10^{-25}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{\frac{1}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 10
Error	1.6
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-29}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 11
Error	4.2
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5999999819625636 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 12
Error	6.6
Cost	941

\[\begin{array}{l} \mathbf{if}\;x \leq -400000:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{elif}\;x \leq -4.999999841327613 \cdot 10^{-21} \lor \neg \left(x \leq 5.000000097707407 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Error	8.4
Cost	480

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

Alternative 14
Error	15.4
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645 \lor \neg \left(x \leq 0.05000000074505806\right):\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15
Error	15.4
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -0.4000000059604645 \lor \neg \left(x \leq 0.05000000074505806\right):\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{2}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Error	15.8
Cost	352

\[\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot 2} \]

Alternative 17
Error	22.7
Cost	224

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

Alternative 18
Error	23.4
Cost	96

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

Reproduce

herbie shell --seed 2023017 
(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))))))