Logistic distribution

Average Error: 0.1 → 0.2

Time: 1.0min

Precision: binary32

Cost: 16480

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

Math

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

↓

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

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 (exp (/ (fabs x) s))) s)
  (pow (+ 1.0 (exp (/ (fabs x) (- s)))) 2.0)))

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) {
	return ((1.0f / expf((fabsf(x) / s))) / s) / powf((1.0f + expf((fabsf(x) / -s))), 2.0f);
}

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
    code = ((1.0e0 / exp((abs(x) / s))) / s) / ((1.0e0 + exp((abs(x) / -s))) ** 2.0e0)
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)
	return Float32(Float32(Float32(Float32(1.0) / exp(Float32(abs(x) / s))) / s) / (Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s)))) ^ Float32(2.0)))
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)
	tmp = ((single(1.0) / exp((abs(x) / s))) / s) / ((single(1.0) + exp((abs(x) / -s))) ^ single(2.0));
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

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational_best-simplify-13 [=>] 0.1	\[ \frac{e^{\frac{\color{blue}{\frac{\left|x\right|}{-1}}}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational_best-simplify-53 [=>] 0.1	\[ \frac{e^{\color{blue}{\frac{\left|x\right|}{-1 \cdot s}}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{\color{blue}{s \cdot -1}}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational_best-simplify-10 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{\color{blue}{-s}}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \color{blue}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right)}} \]
   rational_best-simplify-50 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   rational_best-simplify-3 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   rational_best-simplify-13 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot \left(\left(e^{\frac{\color{blue}{\frac{\left|x\right|}{-1}}}{s}} + 1\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   rational_best-simplify-53 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot \left(\left(e^{\color{blue}{\frac{\left|x\right|}{-1 \cdot s}}} + 1\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot \left(\left(e^{\frac{\left|x\right|}{\color{blue}{s \cdot -1}}} + 1\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   rational_best-simplify-10 [=>] 0.1	\[ \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot \left(\left(e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 1\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

3. Taylor expanded in x around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s \cdot {\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-54 [=>] 0.1	\[ \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}}} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{\frac{e^{\color{blue}{\frac{\left|x\right|}{s} \cdot -1}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-10 [=>] 0.1	\[ \frac{\frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-13 [=>] 0.1	\[ \frac{\frac{e^{\color{blue}{\frac{\frac{\left|x\right|}{s}}{-1}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-54 [<=] 0.1	\[ \frac{\frac{e^{\color{blue}{\frac{\left|x\right|}{s \cdot -1}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-11 [<=] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{\color{blue}{-s}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   rational_best-simplify-1 [=>] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\left(e^{\color{blue}{\frac{\left|x\right|}{s} \cdot -1}} + 1\right)}^{2}} \]
   rational_best-simplify-10 [=>] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\left(e^{\color{blue}{-\frac{\left|x\right|}{s}}} + 1\right)}^{2}} \]
   rational_best-simplify-13 [=>] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\left(e^{\color{blue}{\frac{\frac{\left|x\right|}{s}}{-1}}} + 1\right)}^{2}} \]
   rational_best-simplify-54 [<=] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\left(e^{\color{blue}{\frac{\left|x\right|}{s \cdot -1}}} + 1\right)}^{2}} \]
   rational_best-simplify-11 [<=] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\left(e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 1\right)}^{2}} \]
   rational_best-simplify-3 [=>] 0.1	\[ \frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{{\color{blue}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right)}}^{2}} \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	16448

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

Alternative 2
Error	0.1
Cost	16448

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

Alternative 3
Error	1.2
Cost	13312

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

Alternative 4
Error	1.2
Cost	13248

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

Alternative 5
Error	1.7
Cost	6688

\[\frac{\frac{\frac{1}{e^{\frac{\left|x\right|}{s}}}}{s}}{4} \]

Alternative 6
Error	1.7
Cost	6656

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

Alternative 7
Error	15.9
Cost	3552

\[\frac{\frac{\frac{1}{1 + \frac{\left|x\right|}{s}}}{s}}{4} \]

Alternative 8
Error	23.4
Cost	96

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

Reproduce

herbie shell --seed 2023099 
(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))))))