Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 28.8s

Precision: binary32

Cost: 23136

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := \frac{\left|x\right|}{s}\\ \frac{\frac{e^{t_0}}{s}}{\frac{2}{e^{t_1}} + \left(e^{t_0 - t_1} + 1\right)} \end{array} \]

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 (/ (- (fabs x)) s)) (t_1 (/ (fabs x) s)))
   (/ (/ (exp t_0) s) (+ (/ 2.0 (exp t_1)) (+ (exp (- t_0 t_1)) 1.0)))))

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 = -fabsf(x) / s;
	float t_1 = fabsf(x) / s;
	return (expf(t_0) / s) / ((2.0f / expf(t_1)) + (expf((t_0 - t_1)) + 1.0f));
}

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
    real(4) :: t_1
    t_0 = -abs(x) / s
    t_1 = abs(x) / s
    code = (exp(t_0) / s) / ((2.0e0 / exp(t_1)) + (exp((t_0 - t_1)) + 1.0e0))
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 = Float32(Float32(-abs(x)) / s)
	t_1 = Float32(abs(x) / s)
	return Float32(Float32(exp(t_0) / s) / Float32(Float32(Float32(2.0) / exp(t_1)) + Float32(exp(Float32(t_0 - t_1)) + Float32(1.0))))
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 = -abs(x) / s;
	t_1 = abs(x) / s;
	tmp = (exp(t_0) / s) / ((single(2.0) / exp(t_1)) + (exp((t_0 - t_1)) + single(1.0)));
end

Derivation

1. Initial program 99.7%

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

2. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(\frac{s}{e^{\frac{\left|x\right|}{s}}}, e^{\frac{-\left|x\right|}{s}} + 2, s\right)}} \]
   Step-by-step derivation

   [Start] 99.7%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   /-rgt-identity [<=] 99.7%	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}{1}} \]
   associate-/l/ [=>] 99.7%	\[ \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{1 \cdot \left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   *-lft-identity [=>] 99.7%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   +-commutative [=>] 99.7%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]
   distribute-rgt-in [=>] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{e^{\frac{-\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) + 1 \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   *-lft-identity [=>] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) + \color{blue}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   +-commutative [=>] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) + s \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}} \]
   distribute-rgt-in [=>] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) + \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} \cdot s + 1 \cdot s\right)}} \]
   *-lft-identity [=>] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{e^{\frac{-\left|x\right|}{s}} \cdot \left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) + \left(e^{\frac{-\left|x\right|}{s}} \cdot s + \color{blue}{s}\right)} \]

3. Taylor expanded in s around 0 99.8%

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

4. Simplified 99.8%

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

   [Start] 99.8%	\[ \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s \cdot \left(1 + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right)} \]
   associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{s}}{1 + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   mul-1-neg [=>] 99.8%	\[ \frac{\frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s}}{1 + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)} \]
   +-commutative [=>] 99.8%	\[ \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{s}}{\color{blue}{\left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right) + 1}} \]
   +-commutative [=>] 99.8%	\[ \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{s}}{\color{blue}{\left(2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}} + \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}}\right)} + 1} \]
   associate-+l+ [=>] 99.8%	\[ \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{s}}{\color{blue}{2 \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}} + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 1\right)}} \]
   associate-*r/ [=>] 99.8%	\[ \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{s}}{\color{blue}{\frac{2 \cdot 1}{e^{\frac{\left|x\right|}{s}}}} + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 1\right)} \]
   metadata-eval [=>] 99.8%	\[ \frac{\frac{e^{-\frac{\left|x\right|}{s}}}{s}}{\frac{\color{blue}{2}}{e^{\frac{\left|x\right|}{s}}} + \left(\frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{e^{\frac{\left|x\right|}{s}}} + 1\right)} \]

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	23136

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := \frac{\left|x\right|}{s}\\ \frac{\frac{e^{t_0}}{s}}{\frac{2}{e^{t_1}} + \left(e^{t_0 - t_1} + 1\right)} \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	16448

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

Alternative 3
Accuracy	99.5%
Cost	13248

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

Alternative 4
Accuracy	99.0%
Cost	13248

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

Alternative 5
Accuracy	98.3%
Cost	10184

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)\right)}\\ \mathbf{elif}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + \left(e^{\frac{-x}{s}} + \left(1 + \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot 0.5 - \frac{\left|x\right|}{s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + \left(e^{\frac{-\left|x\right|}{s}} + e^{\frac{x}{s}}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	99.2%
Cost	10148

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \mathbf{if}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + \left(2 + e^{\frac{-x}{s}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + \left(t_0 + e^{\frac{x}{s}}\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	10148

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \mathbf{if}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + \left(t_0 + e^{\frac{-x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + \left(t_0 + e^{\frac{x}{s}}\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	96.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)\right)}\\ \mathbf{elif}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + \left(e^{\frac{-x}{s}} + \left(1 + \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot 0.5 - \frac{\left|x\right|}{s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\mathsf{fma}\left(2, e^{\frac{x}{s}}, 2\right)}\\ \end{array} \]

Alternative 9
Accuracy	95.8%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)\right)}\\ \mathbf{elif}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + \left(e^{\frac{-x}{s}} + \left(1 - \frac{\left|x\right|}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\mathsf{fma}\left(2, e^{\frac{x}{s}}, 2\right)}\\ \end{array} \]

Alternative 10
Accuracy	89.3%
Cost	3688

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00139999995008111:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right) + 4}}{s}\\ \mathbf{elif}\;x \leq -9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 11
Accuracy	84.5%
Cost	941

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00139999995008111:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right) + 4}}{s}\\ \mathbf{elif}\;x \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(x \leq 9.999999998199587 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 12
Accuracy	75.0%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(x \leq 2.00000006274879 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Accuracy	74.1%
Cost	480

\[\frac{\frac{1}{s}}{\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right) + 4} \]

Alternative 14
Accuracy	74.5%
Cost	480

\[\frac{\frac{1}{\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right) + 4}}{s} \]

Alternative 15
Accuracy	58.7%
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Accuracy	58.7%
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 17
Accuracy	57.4%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999682655225 \cdot 10^{-22} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 18
Accuracy	14.9%
Cost	96

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

Reproduce

herbie shell --seed 2023263 
(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))))))