Logistic distribution

Average Error: 0.1 → 0.1

Time: 15.7s

Precision: binary32

Cost: 19840

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

Math

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

↓

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

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_0 (+ s (* s (* t_0 (+ 2.0 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 t_0 / (s + (s * (t_0 * (2.0f + 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 = t_0 / (s + (s * (t_0 * (2.0e0 + 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(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(s + Float32(s * Float32(t_0 * Float32(Float32(2.0) + 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 = t_0 / (s + (s * (t_0 * (single(2.0) + 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

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]
   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_best_oopsla_all_46_json_45_simplify-74 [=>] 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_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s + s \cdot e^{\frac{-\left|x\right|}{s}}\right) + s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right) + \left(s + s \cdot e^{\frac{-\left|x\right|}{s}}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s + \left(s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right) + s \cdot e^{\frac{-\left|x\right|}{s}}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + \color{blue}{\left(s \cdot e^{\frac{-\left|x\right|}{s}} + s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + \left(\color{blue}{e^{\frac{-\left|x\right|}{s}} \cdot s} + s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + \color{blue}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + s \cdot \left(\color{blue}{e^{\frac{-\left|x\right|}{s}} \cdot 1} + e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + s \cdot \left(\color{blue}{1 \cdot e^{\frac{-\left|x\right|}{s}}} + e^{\frac{-\left|x\right|}{s}} \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-37 [<=] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s + s \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} \cdot \left(1 + \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)\right)}} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	19840

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

Alternative 2
Error	0.1
Cost	19840

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

Alternative 3
Error	0.1
Cost	16448

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

Alternative 4
Error	1.6
Cost	13280

\[\begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ 0.5 \cdot \frac{t_0}{s + s \cdot t_0} \end{array} \]

Alternative 5
Error	1.2
Cost	13280

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t_0}{\left(1 + t_0 \cdot 3\right) \cdot s} \end{array} \]

Alternative 6
Error	1.7
Cost	9984

\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left|x\right| \cdot -3 + s \cdot 4} \]

Alternative 7
Error	1.7
Cost	6656

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

Alternative 8
Error	22.6
Cost	3488

\[\frac{1}{s \cdot 4 + \left|x\right| \cdot -2} \]

Alternative 9
Error	23.2
Cost	96

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

Reproduce

herbie shell --seed 2023090 
(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))))))