Logistic distribution

Average Accuracy: 99.5% → 99.6%

Time: 17.7s

Precision: binary32

Cost: 6880

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

Math

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

↓

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

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 (/ x s)))) (/ 1.0 (* s (+ (/ 1.0 t_0) (+ t_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((x / s));
	return 1.0f / (s * ((1.0f / t_0) + (t_0 + 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((x / s))
    code = 1.0e0 / (s * ((1.0e0 / t_0) + (t_0 + 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(x / s))
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(1.0) / t_0) + Float32(t_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((x / s));
	tmp = single(1.0) / (s * ((single(1.0) / t_0) + (t_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%

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

   [Start] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 99.5	\[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*l* [=>] 99.6	\[ \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   times-frac [=>] 99.4	\[ \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-/l* [=>] 99.4	\[ \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]
   distribute-frac-neg [=>] 99.4	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]
   exp-neg [=>] 99.4	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Applied egg-rr 62.9%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\color{blue}{\left(0 + e^{\frac{x}{s}}\right)} + 2\right)} \]
   Proof

   [Start] 99.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)} \]
   add-sqr-sqrt [=>] 99.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\color{blue}{\sqrt{\frac{\left|x\right|}{s}} \cdot \sqrt{\frac{\left|x\right|}{s}}}} + 2\right)} \]
   sqrt-unprod [=>] 99.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\color{blue}{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}}}} + 2\right)} \]
   frac-times [=>] 90.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\sqrt{\color{blue}{\frac{\left|x\right| \cdot \left|x\right|}{s \cdot s}}}} + 2\right)} \]
   sqr-neg [<=] 90.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 2\right)} \]
   frac-times [<=] 99.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\sqrt{\color{blue}{\frac{\left|x\right|}{-s} \cdot \frac{\left|x\right|}{-s}}}} + 2\right)} \]
   sqrt-unprod [<=] -0.0	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\color{blue}{\sqrt{\frac{\left|x\right|}{-s}} \cdot \sqrt{\frac{\left|x\right|}{-s}}}} + 2\right)} \]
   add-sqr-sqrt [<=] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\color{blue}{\frac{\left|x\right|}{-s}}} + 2\right)} \]
   add-log-exp [=>] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\color{blue}{\log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)} + 2\right)} \]
   *-un-lft-identity [=>] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\log \color{blue}{\left(1 \cdot e^{e^{\frac{\left|x\right|}{-s}}}\right)} + 2\right)} \]
   log-prod [=>] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\color{blue}{\left(\log 1 + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right)} + 2\right)} \]
   metadata-eval [=>] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(\color{blue}{0} + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right) + 2\right)} \]
   add-log-exp [<=] 26.5	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + \color{blue}{e^{\frac{\left|x\right|}{-s}}}\right) + 2\right)} \]
   add-sqr-sqrt [=>] 13.1	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{-s}}\right) + 2\right)} \]
   fabs-sqr [=>] 13.1	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-s}}\right) + 2\right)} \]
   add-sqr-sqrt [<=] 63.1	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{\color{blue}{x}}{-s}}\right) + 2\right)} \]
   add-sqr-sqrt [=>] -0.0	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}\right) + 2\right)} \]
   sqrt-unprod [=>] 59.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}\right) + 2\right)} \]
   sqr-neg [=>] 59.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}\right) + 2\right)} \]
   sqrt-unprod [<=] 62.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}\right) + 2\right)} \]
   add-sqr-sqrt [<=] 62.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{s}}}\right) + 2\right)} \]

4. Simplified 62.9%

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

   [Start] 62.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\left(0 + e^{\frac{x}{s}}\right) + 2\right)} \]
   +-lft-identity [=>] 62.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(\color{blue}{e^{\frac{x}{s}}} + 2\right)} \]

5. Taylor expanded in s around 0 63.0%

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

6. Simplified 99.6%

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

   [Start] 63.0	\[ \frac{1}{s \cdot \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   mul-1-neg [=>] 63.0	\[ \frac{1}{s \cdot \left(e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   unpow1 [<=] 63.0	\[ \frac{1}{s \cdot \left(e^{-\frac{\left|\color{blue}{{x}^{1}}\right|}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   sqr-pow [=>] 49.6	\[ \frac{1}{s \cdot \left(e^{-\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   fabs-sqr [=>] 49.6	\[ \frac{1}{s \cdot \left(e^{-\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   sqr-pow [<=] 99.6	\[ \frac{1}{s \cdot \left(e^{-\frac{\color{blue}{{x}^{1}}}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   unpow1 [=>] 99.6	\[ \frac{1}{s \cdot \left(e^{-\frac{\color{blue}{x}}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   distribute-frac-neg [<=] 99.6	\[ \frac{1}{s \cdot \left(e^{\color{blue}{\frac{-x}{s}}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]

7. Applied egg-rr 99.6%

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

   [Start] 99.6	\[ \frac{1}{s \cdot \left(e^{\frac{-x}{s}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   distribute-frac-neg [=>] 99.6	\[ \frac{1}{s \cdot \left(e^{\color{blue}{-\frac{x}{s}}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]
   exp-neg [=>] 99.6	\[ \frac{1}{s \cdot \left(\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(e^{\frac{x}{s}} + 2\right)\right)} \]

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	88.1%
Cost	3628

\[\begin{array}{l} t_0 := \frac{x - x}{s}\\ \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot \left(-s\right)} + t_0\right)}\\ \mathbf{elif}\;x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s \cdot \frac{s}{x}} + t_0\right)}\\ \mathbf{elif}\;x \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{\frac{s \cdot s}{x}} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{e^{\frac{x}{s}}}}{x}\\ \end{array} \]

Alternative 2
Accuracy	95.6%
Cost	3620

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + e^{\frac{x}{s}} \cdot 2\right)}\\ \end{array} \]

Alternative 3
Accuracy	95.6%
Cost	3588

\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}\\ \end{array} \]

Alternative 4
Accuracy	93.1%
Cost	3560

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{elif}\;x \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{\frac{s \cdot s}{x}} + \frac{x - x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{t_0}}{x}\\ \end{array} \]

Alternative 5
Accuracy	93.3%
Cost	3560

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{elif}\;x \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{\frac{s \cdot s}{x}} + \frac{x - x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{t_0}}{x}\\ \end{array} \]

Alternative 6
Accuracy	95.1%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t_0}\\ \end{array} \]

Alternative 7
Accuracy	99.5%
Cost	3552

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

Alternative 8
Accuracy	86.5%
Cost	812

\[\begin{array}{l} t_0 := \frac{x - x}{s}\\ t_1 := \frac{\frac{1}{s}}{4 + \left(\frac{x}{\frac{s \cdot s}{x}} + t_0\right)}\\ \mathbf{if}\;x \leq -2.0000000063421537 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s} \cdot \frac{x}{s} + t_0\right)}\\ \mathbf{elif}\;x \leq 0.004000000189989805:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 9
Accuracy	86.8%
Cost	812

\[\begin{array}{l} t_0 := \frac{x - x}{s}\\ \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot \left(-s\right)} + t_0\right)}\\ \mathbf{elif}\;x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s \cdot \frac{s}{x}} + t_0\right)}\\ \mathbf{elif}\;x \leq 0.004000000189989805:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{\frac{s \cdot s}{x}} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 10
Accuracy	83.2%
Cost	745

\[\begin{array}{l} \mathbf{if}\;x \leq -2.999999892949745 \cdot 10^{-8} \lor \neg \left(x \leq 4.999999969612645 \cdot 10^{-9}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s} \cdot \frac{x}{s} + \frac{x - x}{s}\right)}\\ \end{array} \]

Alternative 11
Accuracy	85.4%
Cost	745

\[\begin{array}{l} \mathbf{if}\;x \leq -2.999999892949745 \cdot 10^{-8} \lor \neg \left(x \leq 4.999999969612645 \cdot 10^{-9}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s \cdot \frac{s}{x}} + \frac{x - x}{s}\right)}\\ \end{array} \]

Alternative 12
Accuracy	84.6%
Cost	652

\[\begin{array}{l} t_0 := \frac{\frac{1}{s}}{4 - \frac{x \cdot x}{s \cdot \left(-s\right)}}\\ \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 0.004000000189989805:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 13
Accuracy	79.3%
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999936531045 \cdot 10^{-19} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Accuracy	62.8%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8} \lor \neg \left(x \leq 4.999999969612645 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15
Accuracy	62.7%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8} \lor \neg \left(x \leq 4.999999969612645 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 16
Accuracy	28.1%
Cost	96

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

Reproduce

herbie shell --seed 2023151 
(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))))))