Logistic distribution

Average Error: 0.2 → 0.2

Time: 16.2s

Precision: binary32

Cost: 6880

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{\frac{-1}{s + \frac{s}{t_0}}}{-1 - t_0} \end{array} \]

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 (/ s t_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 + (s / t_0))) / (-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 + (s / t_0))) / ((-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(Float32(-1.0) / Float32(s + Float32(s / t_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 + (s / t_0))) / (single(-1.0) - t_0);
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 0.2

   \[\leadsto \frac{1}{\color{blue}{\frac{e^{\frac{x}{s}} + 1}{\frac{1}{s + \frac{s}{e^{\frac{x}{s}}}}}}} \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\frac{1}{s + \frac{s}{e^{\frac{x}{s}}}}}{-1 - e^{\frac{x}{s}}} \cdot -1} \]

5. Final simplification 0.2

   \[\leadsto \frac{\frac{-1}{s + \frac{s}{e^{\frac{x}{s}}}}}{-1 - e^{\frac{x}{s}}} \]

Alternatives

Alternative 1
Error	0.2
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 2
Error	0.2
Cost	6880

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{\frac{1}{1 + t_0}}{s + \frac{s}{t_0}} \end{array} \]

Alternative 3
Error	1.2
Cost	6688

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

Alternative 4
Error	1.6
Cost	6656

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

Alternative 5
Error	1.2
Cost	3812

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + e^{\frac{x}{s}}}}{s + \frac{s}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 6
Error	1.4
Cost	3748

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 + e^{\frac{x}{s}}}{\frac{1}{s + \left(s - x\right)}}}\\ \end{array} \]

Alternative 7
Error	1.5
Cost	3652

\[\begin{array}{l} \mathbf{if}\;x \leq 1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + e^{\frac{x}{s}}}}{s + s}\\ \end{array} \]

Alternative 8
Error	1.6
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + t_0 \cdot 2}\\ \end{array} \]

Alternative 9
Error	1.5
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + t_0}}{s + s}\\ \end{array} \]

Alternative 10
Error	1.7
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]

Alternative 11
Error	3.0
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \end{array} \]

Alternative 12
Error	1.7
Cost	3492

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{t_0}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]

Alternative 13
Error	6.2
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8} \lor \neg \left(x \leq 1.999999943436137 \cdot 10^{-9}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 14
Error	6.5
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8} \lor \neg \left(x \leq 1.999999943436137 \cdot 10^{-9}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15
Error	12.0
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7} \lor \neg \left(x \leq 1.999999943436137 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Error	12.0
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 17
Error	23.3
Cost	96

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

Reproduce

herbie shell --seed 2023046 
(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))))))