Logistic distribution

Percentage Accurate: 99.5% → 99.3%

Time: 23.7s

Precision: binary32

Cost: 13316

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \mathbf{if}\;x \leq -4.999999675228202 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(t_0 + 2\right) + e^{-\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\\ \end{array} \]

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)))))
   (if (<= x -4.999999675228202e-39)
     (/ (/ 1.0 s) (+ (+ t_0 2.0) (exp (- (/ x s)))))
     (/ 1.0 (* (+ 1.0 t_0) (fma s (exp (/ x s)) 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) {
	float t_0 = expf((fabsf(x) / -s));
	float tmp;
	if (x <= -4.999999675228202e-39f) {
		tmp = (1.0f / s) / ((t_0 + 2.0f) + expf(-(x / s)));
	} else {
		tmp = 1.0f / ((1.0f + t_0) * fmaf(s, expf((x / s)), s));
	}
	return tmp;
}

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(abs(x) / Float32(-s)))
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999675228202e-39))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(t_0 + Float32(2.0)) + exp(Float32(-Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + t_0) * fma(s, exp(Float32(x / s)), s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999968e-39

   1. Initial program 99.8%

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

   2. Simplified 99.9%

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

      [Start] 99.8%	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
      *-lft-identity [<=] 99.8%	\[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
      associate-*r/ [=>] 99.8%	\[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
      associate-*l* [=>] 99.8%	\[ \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
      times-frac [=>] 99.8%	\[ \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
      associate-*r/ [=>] 99.8%	\[ \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
      associate-/l* [=>] 99.9%	\[ \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]
      distribute-frac-neg [=>] 99.9%	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]
      exp-neg [=>] 99.9%	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      add-sqr-sqrt [=>] 99.8%	\[ \frac{\frac{1}{s}}{e^{\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      sqrt-unprod [=>] 92.7%	\[ \frac{\frac{1}{s}}{e^{\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      sqr-neg [<=] 92.7%	\[ \frac{\frac{1}{s}}{e^{\frac{\sqrt{\color{blue}{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      sqrt-unprod [<=] -0.0%	\[ \frac{\frac{1}{s}}{e^{\frac{\color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      add-sqr-sqrt [<=] 12.1%	\[ \frac{\frac{1}{s}}{e^{\frac{\color{blue}{-\left|x\right|}}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      distribute-frac-neg [=>] 12.1%	\[ \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      exp-neg [=>] 12.1%	\[ \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      add-sqr-sqrt [=>] -0.0%	\[ \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      fabs-sqr [=>] -0.0%	\[ \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      add-sqr-sqrt [<=] 99.9%	\[ \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

   4. Simplified 99.9%

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

      [Start] 99.9%	\[ \frac{\frac{1}{s}}{\frac{1}{e^{\frac{x}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      rec-exp [=>] 99.9%	\[ \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
      distribute-neg-frac [=>] 99.9%	\[ \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

   if -4.99999968e-39 < x

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

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

   3. Applied egg-rr 99.8%

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

      [Start] 99.8%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]
      *-un-lft-identity [=>] 99.8%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, \color{blue}{1 \cdot e^{\frac{\left|x\right|}{s}}}, s\right)} \]
      add-sqr-sqrt [=>] 99.7%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, 1 \cdot e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}, s\right)} \]
      fabs-sqr [=>] 99.7%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, 1 \cdot e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}, s\right)} \]
      add-sqr-sqrt [<=] 99.8%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, 1 \cdot e^{\frac{\color{blue}{x}}{s}}, s\right)} \]

   4. Simplified 99.8%

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

      [Start] 99.8%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, 1 \cdot e^{\frac{x}{s}}, s\right)} \]
      *-lft-identity [=>] 99.8%	\[ \frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, \color{blue}{e^{\frac{x}{s}}}, s\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999675228202 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(e^{\frac{\left|x\right|}{-s}} + 2\right) + e^{-\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13316

\[\begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ \mathbf{if}\;x \leq -4.999999675228202 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(t_0 + 2\right) + e^{-\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)}\\ \end{array} \]

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	99.0%
Cost	13248

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

Alternative 4
Accuracy	99.1%
Cost	10020

\[\begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \left(\frac{1}{\frac{s}{x} \cdot \frac{s}{x}} + \frac{x \cdot 2}{s}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	95.1%
Cost	6656

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

Alternative 6
Accuracy	95.8%
Cost	4004

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ t_1 := 1 + \left(1 + \frac{x}{s}\right)\\ \mathbf{if}\;x \leq 5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;t_0 \cdot \frac{1}{s \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot t_0\right)}\\ \end{array} \]

Alternative 7
Accuracy	95.1%
Cost	3620

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;t_0 \cdot \frac{1}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot t_0\right)}\\ \end{array} \]

Alternative 8
Accuracy	88.9%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 6.380000008194408 \cdot 10^{-27}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{1}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9
Accuracy	89.0%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 6.380000008194408 \cdot 10^{-27}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 10
Accuracy	88.3%
Cost	3428

\[\begin{array}{l} \mathbf{if}\;x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{x \cdot x}{s \cdot s} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	87.2%
Cost	3364

\[\begin{array}{l} \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{x \cdot x}{s \cdot s} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	79.9%
Cost	804

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

Alternative 13
Accuracy	79.9%
Cost	676

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

Alternative 14
Accuracy	84.5%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000156871975 \cdot 10^{-23} \lor \neg \left(x \leq 9.999999998199587 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15
Accuracy	56.6%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -7.99999999855967 \cdot 10^{-23} \lor \neg \left(x \leq 4.999999980020986 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Accuracy	14.5%
Cost	96

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

Reproduce

herbie shell --seed 2023255 
(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))))))