Logistic distribution

Average Error: 0.1 → 0.1

Time: 18.2s

Precision: binary32

Cost: 19840

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

Math

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

↓

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

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_1 (+ t_0 1.0)))
   (/ t_0 (* s (* t_1 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 / (s * (t_1 * 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 / (s * (t_1 * 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(t_0 / Float32(s * Float32(t_1 * 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 / (s * (t_1 * t_1));
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

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

   [Start] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*l* [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   +-commutative [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   +-commutative [=>] 0.1	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	10080

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

Alternative 2
Error	0.1
Cost	6880

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

Alternative 3
Error	1.5
Cost	6688

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

Alternative 4
Error	1.6
Cost	6656

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

Alternative 5
Error	1.3
Cost	4196

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{\left(1 + \frac{1}{\frac{s}{x}}\right) + \frac{0.5}{\frac{s}{\frac{x}{\frac{s}{x}}}}}\right)}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	4132

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 4.000000094968912 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{\left(1 + \frac{x}{s}\right) + 0.5 \cdot \frac{x}{\frac{s \cdot s}{x}}}\right)}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.999999964390316 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{1 + \frac{x}{s}}\right)}\\ \end{array} \]

Alternative 8
Error	1.1
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.999999964390316 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{-1 - t_0}}{s + \frac{s}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 9
Error	1.4
Cost	3748

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 4.000000094968912 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(\frac{x}{s} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + t_0\right)}\\ \end{array} \]

Alternative 10
Error	1.9
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + t_0\right)}\\ \end{array} \]

Alternative 11
Error	1.5
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.999999964390316 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + t_0\right)}\\ \end{array} \]

Alternative 12
Error	2.2
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 4.000000094968912 \cdot 10^{-33}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{0.25}{s}\\ \mathbf{elif}\;x \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(\frac{x}{s} + 2\right) + 0.5 \cdot \frac{x}{\frac{s \cdot s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	3.1
Cost	812

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.000000094968912 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 - \frac{x}{\frac{s}{x} \cdot \left(-s\right)}}\\ \mathbf{elif}\;x \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\left(\frac{x}{s} + 2\right) + 0.5 \cdot \frac{x}{\frac{s \cdot s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Error	3.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 - \frac{x}{\frac{s}{x} \cdot \left(-s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Error	4.2
Cost	552

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Error	4.6
Cost	232

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 17
Error	8.3
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022354 
(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))))))