Logistic distribution

Average Accuracy: 99.5% → 99.5%

Time: 17.5s

Precision: binary32

Cost: 10116

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

Math

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

↓

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

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 (/ (- x) s))))
   (if (<= x -1.0000000359391298e-36)
     (/ (/ 1.0 s) (+ (exp (/ (fabs x) s)) (+ 2.0 (exp (/ x s)))))
     (/ (/ t_0 s) (pow (+ 1.0 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));
	float tmp;
	if (x <= -1.0000000359391298e-36f) {
		tmp = (1.0f / s) / (expf((fabsf(x) / s)) + (2.0f + expf((x / s))));
	} else {
		tmp = (t_0 / s) / powf((1.0f + t_0), 2.0f);
	}
	return tmp;
}

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) :: tmp
    t_0 = exp((-x / s))
    if (x <= (-1.0000000359391298e-36)) then
        tmp = (1.0e0 / s) / (exp((abs(x) / s)) + (2.0e0 + exp((x / s))))
    else
        tmp = (t_0 / s) / ((1.0e0 + t_0) ** 2.0e0)
    end if
    code = tmp
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(-x) / s))
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000359391298e-36))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / s)) + Float32(Float32(2.0) + exp(Float32(x / s)))));
	else
		tmp = Float32(Float32(t_0 / s) / (Float32(Float32(1.0) + t_0) ^ Float32(2.0)));
	end
	return tmp
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_2 = code(x, s)
	t_0 = exp((-x / s));
	tmp = single(0.0);
	if (x <= single(-1.0000000359391298e-36))
		tmp = (single(1.0) / s) / (exp((abs(x) / s)) + (single(2.0) + exp((x / s))));
	else
		tmp = (t_0 / s) / ((single(1.0) + t_0) ^ single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000004e-36

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

   3. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{\left(0 + e^{\frac{x}{s}}\right)} + 2\right)} \]
      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-log-exp [=>] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{\log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)} + 2\right)} \]
      *-un-lft-identity [=>] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\log \color{blue}{\left(1 \cdot e^{e^{\frac{\left|x\right|}{-s}}}\right)} + 2\right)} \]
      log-prod [=>] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{\left(\log 1 + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right)} + 2\right)} \]
      metadata-eval [=>] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(\color{blue}{0} + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right) + 2\right)} \]
      add-log-exp [<=] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + \color{blue}{e^{\frac{\left|x\right|}{-s}}}\right) + 2\right)} \]
      add-sqr-sqrt [=>] -0.0	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{-s}}\right) + 2\right)} \]
      fabs-sqr [=>] -0.0	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-s}}\right) + 2\right)} \]
      add-sqr-sqrt [<=] 95.2	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{\color{blue}{x}}{-s}}\right) + 2\right)} \]
      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 [=>] 98.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}\right) + 2\right)} \]
      sqr-neg [=>] 98.4	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}\right) + 2\right)} \]
      sqrt-unprod [<=] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}\right) + 2\right)} \]
      add-sqr-sqrt [<=] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{x}{\color{blue}{s}}}\right) + 2\right)} \]

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\left(0 + e^{\frac{x}{s}}\right) + 2\right)} \]
      +-lft-identity [=>] 99.9	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(\color{blue}{e^{\frac{x}{s}}} + 2\right)} \]

   if -1.00000004e-36 < 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. Taylor expanded in x around 0 99.8%

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

   3. Simplified 99.8%

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

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

   4. Applied egg-rr 99.7%

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

      [Start] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}^{2}} \]
      distribute-frac-neg [=>] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{-\frac{\left|x\right|}{s}}} + 1\right)}^{2}} \]
      exp-neg [=>] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + 1\right)}^{2}} \]
      add-sqr-sqrt [=>] 98.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}} + 1\right)}^{2}} \]
      fabs-sqr [=>] 98.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + 1\right)}^{2}} \]
      add-sqr-sqrt [<=] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + 1\right)}^{2}} \]

   5. Simplified 99.7%

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

      [Start] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{x}{s}}} + 1\right)}^{2}} \]
      rec-exp [=>] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\color{blue}{e^{-\frac{x}{s}}} + 1\right)}^{2}} \]
      distribute-neg-frac [=>] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{\frac{-x}{s}}} + 1\right)}^{2}} \]

   6. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}^{2}} \]
      distribute-frac-neg [=>] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{-\frac{\left|x\right|}{s}}} + 1\right)}^{2}} \]
      exp-neg [=>] 99.8	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + 1\right)}^{2}} \]
      add-sqr-sqrt [=>] 98.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}} + 1\right)}^{2}} \]
      fabs-sqr [=>] 98.2	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + 1\right)}^{2}} \]
      add-sqr-sqrt [<=] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + 1\right)}^{2}} \]

   7. Simplified 99.8%

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

      [Start] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\frac{1}{e^{\frac{x}{s}}} + 1\right)}^{2}} \]
      rec-exp [=>] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(\color{blue}{e^{-\frac{x}{s}}} + 1\right)}^{2}} \]
      distribute-neg-frac [=>] 99.7	\[ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(e^{\color{blue}{\frac{-x}{s}}} + 1\right)}^{2}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	19840

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

Alternative 2
Accuracy	99.6%
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	99.6%
Cost	13312

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

Alternative 4
Accuracy	99.4%
Cost	13248

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

Alternative 5
Accuracy	99.2%
Cost	10020

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\frac{1}{s}}{t_0 + 3}}\right)}^{3}\\ \end{array} \]

Alternative 6
Accuracy	96.3%
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	94.7%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq -3.000000085396614 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{e^{\frac{x}{s}}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{s}{e^{\frac{-x}{s}}}}\\ \end{array} \]

Alternative 9
Accuracy	95.6%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.1400000212403256 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{t_0}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]

Alternative 10
Accuracy	90.8%
Cost	3524

\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999918644759 \cdot 10^{-31}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{s}{e^{\frac{-x}{s}}}}\\ \end{array} \]

Alternative 11
Accuracy	74.9%
Cost	3428

\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999918644759 \cdot 10^{-31}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s}\\ \mathbf{elif}\;x \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25 + \frac{-0.0625}{\frac{s}{x} \cdot \frac{s}{x}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 12
Accuracy	62.6%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6} \lor \neg \left(x \leq 1.999999943436137 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.0625}{\frac{s}{x} \cdot \frac{s}{x}}}{s}\\ \end{array} \]

Alternative 13
Accuracy	62.5%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6} \lor \neg \left(x \leq 1.999999943436137 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Accuracy	26.7%
Cost	96

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

Reproduce

herbie shell --seed 2023157 
(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))))))