Logistic distribution

Average Accuracy: 99.5% → 99.5%

Time: 19.7s

Precision: binary32

Cost: 16448

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{\frac{t_0}{s}}{{\left(t_0 + 1\right)}^{2}} \end{array} \]

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) (pow (+ t_0 1.0) 2.0))))

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) / powf((t_0 + 1.0f), 2.0f);
}

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
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = (t_0 / s) / ((t_0 + 1.0e0) ** 2.0e0)
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)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(Float32(t_0 / s) / (Float32(t_0 + Float32(1.0)) ^ Float32(2.0)))
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) / ((t_0 + single(1.0)) ^ single(2.0));
end

Derivation

1. Initial program 99.5%

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

2. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*l* [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   +-commutative [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   +-commutative [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Taylor expanded in x around 0 99.5%

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

4. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}^{2}}} \]
   Proof

   [Start] 99.5	\[ \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s \cdot {\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   associate-/r* [=>] 99.5	\[ \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}}} \]
   mul-1-neg [=>] 99.5	\[ \frac{\frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   distribute-neg-frac [=>] 99.5	\[ \frac{\frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{s}}{{\left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
   mul-1-neg [=>] 99.5	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{-\frac{\left|x\right|}{s}}} + 1\right)}^{2}} \]
   distribute-neg-frac [=>] 99.5	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{\frac{-\left|x\right|}{s}}} + 1\right)}^{2}} \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13184

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{e^{-\mathsf{log1p}\left(t_0\right)}}{s + \frac{s}{t_0}} \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	6944

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

Alternative 3
Accuracy	99.5%
Cost	6880

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

Alternative 4
Accuracy	99.5%
Cost	6880

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

Alternative 5
Accuracy	99.5%
Cost	6880

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

Alternative 6
Accuracy	96.2%
Cost	6688

\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]

Alternative 7
Accuracy	94.8%
Cost	6656

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

Alternative 8
Accuracy	95.8%
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s + \frac{s}{1 + \frac{x}{s}}}}{1 + t_0}\\ \end{array} \]

Alternative 9
Accuracy	95.5%
Cost	3684

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s + \left(s - x\right)}}{1 + t_0}\\ \end{array} \]

Alternative 10
Accuracy	87.1%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + \left(\frac{x}{s} + \left(1 + \frac{x \cdot x}{\frac{s \cdot s}{0.5}}\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	90.6%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.2000000284906736 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(1 + t_0\right)}\\ \end{array} \]

Alternative 12
Accuracy	95.1%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -2.0000004306106665 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t_0}\\ \end{array} \]

Alternative 13
Accuracy	75.2%
Cost	740

\[\begin{array}{l} \mathbf{if}\;x \leq -39999999311872:\\ \;\;\;\;\frac{1}{2 \cdot \left(s \cdot 2 + \left(0.5 \cdot \left(x \cdot \frac{x}{s}\right) - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + \left(\frac{x}{s} + \left(1 + \frac{x \cdot x}{\frac{s \cdot s}{0.5}}\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	75.1%
Cost	676

\[\begin{array}{l} \mathbf{if}\;x \leq -39999999311872:\\ \;\;\;\;\frac{1}{2 \cdot \left(s \cdot 2 + \left(0.5 \cdot \left(x \cdot \frac{x}{s}\right) - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{2 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)}\\ \end{array} \]

Alternative 15
Accuracy	63.1%
Cost	544

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

Alternative 16
Accuracy	50.7%
Cost	416

\[\frac{1}{s} \cdot \frac{1}{\frac{2 + \frac{x}{s}}{0.5}} \]

Alternative 17
Accuracy	51.3%
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806 \lor \neg \left(x \leq 4.999999980020986 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\frac{0.5}{s}}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 18
Accuracy	50.5%
Cost	288

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

Alternative 19
Accuracy	26.6%
Cost	96

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

Reproduce

herbie shell --seed 2023138 
(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))))))