Logistic distribution

Average Error: 0.2 → 0.2

Time: 18.7s

Precision: binary32

Cost: 13312

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

Math

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

↓

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

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 (- -1.0 (exp (/ (fabs x) s))))
  (* s (+ (exp (/ (- (fabs x)) s)) 1.0))))

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 / (-1.0f - expf((fabsf(x) / s)))) / (s * (expf((-fabsf(x) / s)) + 1.0f));
}

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) / ((-1.0e0) - exp((abs(x) / s)))) / (s * (exp((-abs(x) / s)) + 1.0e0))
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(-1.0) / Float32(Float32(-1.0) - exp(Float32(abs(x) / s)))) / Float32(s * Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(1.0))))
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) / (single(-1.0) - exp((abs(x) / s)))) / (s * (exp((-abs(x) / s)) + single(1.0)));
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

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

   [Start] 0.2	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-46 [=>] 0.2	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}{1 + e^{\frac{-\left|x\right|}{s}}}} \]
   rational.json-simplify-44 [=>] 0.2	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   rational.json-simplify-1 [=>] 0.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{e^{\frac{-\left|x\right|}{s}} + 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-1 [=>] 0.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} + 1}}{s \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]

3. Applied egg-rr 0.2

   \[\leadsto \frac{\color{blue}{\frac{1}{1 + e^{-\frac{\left|x\right|}{-s}}} + 0}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]

4. Simplified 0.2

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

   [Start] 0.2	\[ \frac{\frac{1}{1 + e^{-\frac{\left|x\right|}{-s}}} + 0}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-4 [=>] 0.2	\[ \frac{\color{blue}{\frac{1}{1 + e^{-\frac{\left|x\right|}{-s}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-17 [=>] 0.2	\[ \frac{\frac{1}{\color{blue}{e^{-\frac{\left|x\right|}{-s}} - -1}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-50 [=>] 0.2	\[ \frac{\color{blue}{\frac{-1}{-1 - e^{-\frac{\left|x\right|}{-s}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{\color{blue}{-1}}{-1 - e^{-\frac{\left|x\right|}{-s}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-10 [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\color{blue}{\frac{\frac{\left|x\right|}{-s}}{-1}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-8 [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\frac{\frac{\left|x\right|}{\color{blue}{s \cdot -1}}}{-1}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-46 [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\frac{\color{blue}{\frac{\frac{\left|x\right|}{s}}{-1}}}{-1}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-47 [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\color{blue}{\frac{\frac{\left|x\right|}{s}}{-1 \cdot -1}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\frac{\frac{\left|x\right|}{s}}{\color{blue}{1}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]
   rational.json-simplify-7 [=>] 0.2	\[ \frac{\frac{-1}{-1 - e^{\color{blue}{\frac{\left|x\right|}{s}}}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \]

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.6
Cost	13280

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

Alternative 2
Error	1.6
Cost	6752

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

Alternative 3
Error	1.6
Cost	6688

\[\frac{0.5}{s \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]

Alternative 4
Error	1.8
Cost	6656

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

Alternative 5
Error	1.7
Cost	6656

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

Alternative 6
Error	22.4
Cost	6564

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00018000000272877514:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\left|x\right|}\\ \end{array} \]

Alternative 7
Error	15.8
Cost	3552

\[\frac{\frac{-1}{-2 - \frac{\left|x\right|}{s}}}{s + s} \]

Alternative 8
Error	22.9
Cost	3520

\[\frac{1}{\left(s + s\right) + \left(-\left|x\right|\right)} \cdot 0.5 \]

Alternative 9
Error	22.9
Cost	3456

\[\frac{0.5}{s \cdot 2 + \left(-\left|x\right|\right)} \]

Alternative 10
Error	23.4
Cost	96

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

Reproduce

herbie shell --seed 2023074 
(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))))))