Logistic distribution

Average Error: 0.2 → 0.1

Time: 10.1s

Precision: binary32

Cost: 19840

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t_0 + 1\\ \frac{\frac{t_0}{t_1}}{s \cdot t_1} \end{array} \]

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))) (t_1 (+ t_0 1.0)))
   (/ (/ t_0 t_1) (* s t_1))))

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));
	float t_1 = t_0 + 1.0f;
	return (t_0 / t_1) / (s * t_1);
}

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
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    code = (t_0 / t_1) / (s * t_1)
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(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 + Float32(1.0))
	return Float32(Float32(t_0 / t_1) / Float32(s * t_1))
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));
	t_1 = t_0 + single(1.0);
	tmp = (t_0 / t_1) / (s * t_1);
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 0.1

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

   [Start] 0.2	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 [=>] 0.1	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   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. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	13312

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

Alternative 2
Error	0.2
Cost	13312

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

Alternative 3
Error	1.6
Cost	6752

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

Alternative 4
Error	1.7
Cost	6720

\[\frac{\frac{\frac{e^{\frac{\left|x\right|}{-s}}}{s}}{2}}{2} \]

Alternative 5
Error	22.3
Cost	6564

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \]

Alternative 6
Error	16.0
Cost	3584

\[\frac{\frac{1}{s}}{\frac{\left|x\right|}{-s} + 2} \cdot 0.5 \]

Alternative 7
Error	22.6
Cost	3424

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

Alternative 8
Error	22.6
Cost	3424

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

Alternative 9
Error	23.3
Cost	96

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

Reproduce

herbie shell --seed 2023075 
(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))))))