Logistic distribution

Average Error: 0.1 → 0.1

Time: 17.0s

Precision: binary32

Cost: 13344

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

Math

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

↓

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

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 (/ (fabs x) (- s)))))
   (/ (/ 1.0 (+ 1.0 t_0)) (+ s (/ s 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((fabsf(x) / -s));
	return (1.0f / (1.0f + t_0)) / (s + (s / 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((abs(x) / -s))
    code = (1.0e0 / (1.0e0 + t_0)) / (s + (s / 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(abs(x) / Float32(-s)))
	return Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + t_0)) / Float32(s + Float32(s / 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((abs(x) / -s));
	tmp = (single(1.0) / (single(1.0) + t_0)) / (s + (s / 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

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-46 [=>] 0.1	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}{1 + e^{\frac{-\left|x\right|}{s}}}} \]
   rational.json-simplify-44 [=>] 0.1	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{e^{\frac{-\left|x\right|}{s}} + 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} + 1}}{s \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{-s}}}\right)} + 0 \]
   rational.json-simplify-4 [=>] 0.1	\[ \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{-s}}}\right)}} \]
   rational.json-simplify-46 [=>] 0.1	\[ \color{blue}{\frac{\frac{1}{e^{\frac{\left|x\right|}{-s}} + 1}}{s + \frac{s}{e^{\frac{\left|x\right|}{-s}}}}} \]
   rational.json-simplify-1 [=>] 0.1	\[ \frac{\frac{1}{\color{blue}{1 + e^{\frac{\left|x\right|}{-s}}}}}{s + \frac{s}{e^{\frac{\left|x\right|}{-s}}}} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13344

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

Alternative 2
Error	0.1
Cost	13312

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

Alternative 3
Error	0.1
Cost	13312

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

Alternative 4
Error	1.5
Cost	6720

\[\frac{0.5}{s + \frac{s}{e^{\frac{\left|x\right|}{-s}}}} \]

Alternative 5
Error	1.5
Cost	6688

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

Alternative 6
Error	1.7
Cost	6656

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

Alternative 7
Error	1.6
Cost	6656

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

Alternative 8
Error	22.6
Cost	3424

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

Alternative 9
Error	22.6
Cost	3424

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

Alternative 10
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))))))