Logistic distribution

Average Accuracy: 99.5% → 99.5%

Time: 19.5s

Precision: binary32

Cost: 6912

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

Math

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

↓

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

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
 (/ 1.0 (* (* s (+ 1.0 (exp (/ x s)))) (+ 1.0 (exp (/ (- x) s))))))

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) {
	return 1.0f / ((s * (1.0f + expf((x / s)))) * (1.0f + expf((-x / s))));
}

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
    code = 1.0e0 / ((s * (1.0e0 + exp((x / s)))) * (1.0e0 + exp((-x / s))))
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)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(Float32(1.0) + exp(Float32(x / s)))) * Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))))
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)
	tmp = single(1.0) / ((s * (single(1.0) + exp((x / s)))) * (single(1.0) + exp((-x / s))));
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%

   \[\leadsto \color{blue}{\frac{1}{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   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	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 99.5	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 99.5	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 99.5	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 99.5	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l* [=>] 99.4	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 99.5	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Applied egg-rr 97.5%

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}} \]
   Proof

   [Start] 99.5	\[ \frac{1}{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]
   add-exp-log [=>] 97.7	\[ \frac{1}{\color{blue}{e^{\log \left(\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)\right)}}} \]
   *-commutative [=>] 97.7	\[ \frac{1}{e^{\log \color{blue}{\left(\left(1 + e^{\frac{\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)\right)}}} \]
   log-prod [=>] 97.4	\[ \frac{1}{e^{\color{blue}{\log \left(1 + e^{\frac{\left|x\right|}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}}} \]
   log1p-def [=>] 97.4	\[ \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{s}}\right)} + \log \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   add-sqr-sqrt [=>] 48.6	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   fabs-sqr [=>] 48.6	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   add-sqr-sqrt [<=] 60.9	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{\color{blue}{x}}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   add-sqr-sqrt [=>] 48.6	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}}\right)}} \]
   fabs-sqr [=>] 48.6	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}\right)}} \]
   add-sqr-sqrt [<=] 97.5	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{\color{blue}{x}}{s}}}\right)}} \]

4. Simplified 99.6%

   \[\leadsto \frac{1}{\color{blue}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}} \]
   Proof

   [Start] 97.5	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}} \]
   exp-sum [=>] 97.7	\[ \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot e^{\log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}} \]
   rem-exp-log [=>] 99.6	\[ \frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot \color{blue}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right)}} \]
   *-commutative [=>] 99.6	\[ \frac{1}{\color{blue}{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}} \]

5. Taylor expanded in s around 0 99.5%

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

6. Simplified 99.5%

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

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

7. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.6%
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 2
Accuracy	96.1%
Cost	6688

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

Alternative 3
Accuracy	94.7%
Cost	6656

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

Alternative 4
Accuracy	95.7%
Cost	4004

\[\begin{array}{l} t_0 := 1 + e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.0000000095132306 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{1}{s + \frac{s}{t_0 + -1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{1}{s + \left(\left(s - x\right) + 0.5 \cdot \frac{x}{\frac{s}{x}}\right)}}}\\ \end{array} \]

Alternative 5
Accuracy	95.4%
Cost	3812

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{1}{s + \frac{s}{\left(1 + t_0\right) + -1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]

Alternative 6
Accuracy	77.9%
Cost	3692

\[\begin{array}{l} t_0 := \frac{1}{s + s \cdot \left(\left(1 + \frac{x}{s}\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \cdot \left(0.5 + \frac{x}{s} \cdot 0.25\right)\\ \mathbf{if}\;x \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{1}{s + \left(s + \left(0.5 \cdot \left(\frac{1}{s} \cdot \left(x \cdot x\right)\right) - x\right)\right)}}}\\ \mathbf{elif}\;x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, \frac{x}{\frac{s}{x}}\right)}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} + -2\right)}\\ \end{array} \]

Alternative 7
Accuracy	86.5%
Cost	3624

\[\begin{array}{l} \mathbf{if}\;x \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{1}{s + \left(s + \left(0.5 \cdot \left(\frac{1}{s} \cdot \left(x \cdot x\right)\right) - x\right)\right)}}}\\ \mathbf{elif}\;x \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s + s \cdot \left(\left(1 + \frac{x}{s}\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \cdot \left(0.5 + \frac{x}{s} \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 8
Accuracy	95.5%
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	1200

\[\begin{array}{l} t_0 := \frac{1}{s + s \cdot \left(\left(1 + \frac{x}{s}\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \cdot \left(0.5 + \frac{x}{s} \cdot 0.25\right)\\ \mathbf{if}\;x \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{1}{s + \left(s + \left(0.5 \cdot \left(\frac{1}{s} \cdot \left(x \cdot x\right)\right) - x\right)\right)}}}\\ \mathbf{elif}\;x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 50000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} + -2\right)}\\ \end{array} \]

Alternative 10
Accuracy	63.1%
Cost	608

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

Alternative 11
Accuracy	64.1%
Cost	488

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} + -2\right)}\\ \mathbf{elif}\;x \leq 3.5000000934815034 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{2 \cdot \frac{s}{x \cdot x}}}\\ \end{array} \]

Alternative 12
Accuracy	64.1%
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999494757503 \cdot 10^{-5} \lor \neg \left(x \leq 3.5000000934815034 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{x}{s} + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Accuracy	63.0%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999494757503 \cdot 10^{-5} \lor \neg \left(x \leq 3.5000000934815034 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Accuracy	8.8%
Cost	96

\[\frac{-0.5}{x} \]

Alternative 15
Accuracy	27.4%
Cost	96

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

Reproduce

herbie shell --seed 2023137 
(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))))))