Logistic distribution

Average Error: 0.1 → 0.1

Time: 15.9s

Precision: binary32

Cost: 16512

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

Math

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

↓

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

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
 (/
  (exp (/ (- (fabs x)) s))
  (/ s (/ 1.0 (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 expf((-fabsf(x) / s)) / (s / (1.0f / 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 = exp((-abs(x) / s)) / (s / (1.0e0 / ((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(exp(Float32(Float32(-abs(x)) / s)) / Float32(s / Float32(Float32(1.0) / (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 = exp((-abs(x) / s)) / (s / (single(1.0) / ((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

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]
   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.json-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]
   rational.json-simplify-56 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} - -1\right)}} \]
   rational.json-simplify-47 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(-s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(-1 - e^{\frac{-\left|x\right|}{s}}\right)}} \]
   rational.json-simplify-31 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot -1\right)} \cdot \left(-1 - e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-31 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(-s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot \left(-1 - e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-47 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} - -1\right)}} \]
   rational.json-simplify-56 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]
   rational.json-simplify-1 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

3. Applied egg-rr 0.1

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

4. Taylor expanded in x around 0 0.1

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

5. Simplified 0.1

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\frac{s}{\frac{1}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}}}} \]
   rational.json-simplify-1 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\frac{s}{\frac{1}{{\color{blue}{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}}^{2}}}} \]
   rational.json-simplify-41 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\frac{s}{\frac{1}{{\left(1 + e^{\color{blue}{\frac{-1}{\frac{s}{\left|x\right|}}}}\right)}^{2}}}} \]
   rational.json-simplify-13 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\frac{s}{\frac{1}{{\left(1 + e^{\color{blue}{\frac{\left|x\right|}{\frac{s}{-1}}}}\right)}^{2}}}} \]
   rational.json-simplify-17 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\frac{s}{\frac{1}{{\left(1 + e^{\frac{\left|x\right|}{\color{blue}{-s}}}\right)}^{2}}}} \]

6. Final simplification 0.1

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

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	1.2
Cost	13248

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

Alternative 3
Error	1.7
Cost	6656

\[\frac{0.25}{s} \cdot e^{\frac{\left|x\right|}{-s}} \]

Alternative 4
Error	1.6
Cost	6656

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

Alternative 5
Error	23.3
Cost	96

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

Reproduce

herbie shell --seed 2023073 
(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))))))