Logistic distribution

Average Accuracy: 99.5% → 99.5%

Time: 16.5s

Precision: binary32

Cost: 6880

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

Math

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

↓

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

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 (fma 2.0 (/ 1.0 (/ 2.0 (* 2.0 (cosh (/ x s))))) 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) {
	return 1.0f / (s * fmaf(2.0f, (1.0f / (2.0f / (2.0f * coshf((x / s))))), 2.0f));
}

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(s * fma(Float32(2.0), Float32(Float32(1.0) / Float32(Float32(2.0) / Float32(Float32(2.0) * cosh(Float32(x / s))))), Float32(2.0))))
end

Derivation

1. Initial program 99.6%

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

2. Simplified 99.3%

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

   [Start] 99.6	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 99.6	\[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   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 [=>] 99.2	\[ \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.2	\[ \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-/l* [=>] 99.2	\[ \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]
   distribute-frac-neg [=>] 99.2	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]
   exp-neg [=>] 99.2	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Applied egg-rr 60.5%

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

   [Start] 99.3	\[ \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [=>] 99.2	\[ \frac{\frac{1}{s}}{e^{\color{blue}{\sqrt{\frac{\left|x\right|}{s}} \cdot \sqrt{\frac{\left|x\right|}{s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqrt-unprod [=>] 99.3	\[ \frac{\frac{1}{s}}{e^{\color{blue}{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\left|x\right|}{s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   frac-times [=>] 90.3	\[ \frac{\frac{1}{s}}{e^{\sqrt{\color{blue}{\frac{\left|x\right| \cdot \left|x\right|}{s \cdot s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqr-neg [<=] 90.3	\[ \frac{\frac{1}{s}}{e^{\sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   frac-times [<=] 99.3	\[ \frac{\frac{1}{s}}{e^{\sqrt{\color{blue}{\frac{\left|x\right|}{-s} \cdot \frac{\left|x\right|}{-s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqrt-unprod [<=] -0.0	\[ \frac{\frac{1}{s}}{e^{\color{blue}{\sqrt{\frac{\left|x\right|}{-s}} \cdot \sqrt{\frac{\left|x\right|}{-s}}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [<=] 23.7	\[ \frac{\frac{1}{s}}{e^{\color{blue}{\frac{\left|x\right|}{-s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-log-exp [=>] 23.7	\[ \frac{\frac{1}{s}}{\color{blue}{\log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   *-un-lft-identity [=>] 23.7	\[ \frac{\frac{1}{s}}{\log \color{blue}{\left(1 \cdot e^{e^{\frac{\left|x\right|}{-s}}}\right)} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   log-prod [=>] 23.7	\[ \frac{\frac{1}{s}}{\color{blue}{\left(\log 1 + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right)} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   metadata-eval [=>] 23.7	\[ \frac{\frac{1}{s}}{\left(\color{blue}{0} + \log \left(e^{e^{\frac{\left|x\right|}{-s}}}\right)\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-log-exp [<=] 23.7	\[ \frac{\frac{1}{s}}{\left(0 + \color{blue}{e^{\frac{\left|x\right|}{-s}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [=>] 11.4	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{-s}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   fabs-sqr [=>] 11.4	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-s}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [<=] 62.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{\color{blue}{x}}{-s}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [=>] -0.0	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqrt-unprod [=>] 57.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqr-neg [=>] 57.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   sqrt-unprod [<=] 60.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   add-sqr-sqrt [<=] 60.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{\color{blue}{s}}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

4. Simplified 60.5%

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

   [Start] 60.5	\[ \frac{\frac{1}{s}}{\left(0 + e^{\frac{x}{s}}\right) + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
   +-lft-identity [=>] 60.5	\[ \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

5. Taylor expanded in s around 0 60.5%

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

6. Simplified 99.3%

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

   [Start] 60.5	\[ \frac{1}{s \cdot \left(e^{\frac{x}{s}} + \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 2\right)\right)} \]
   associate-/r* [=>] 60.5	\[ \color{blue}{\frac{\frac{1}{s}}{e^{\frac{x}{s}} + \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + 2\right)}} \]
   associate-+r+ [=>] 60.5	\[ \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + e^{-1 \cdot \frac{\left|x\right|}{s}}\right) + 2}} \]
   +-commutative [=>] 60.5	\[ \frac{\frac{1}{s}}{\color{blue}{2 + \left(e^{\frac{x}{s}} + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}} \]
   mul-1-neg [=>] 60.5	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right)} \]
   unpow1 [<=] 60.5	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{-\frac{\left|\color{blue}{{x}^{1}}\right|}{s}}\right)} \]
   sqr-pow [=>] 48.2	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{-\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}\right)} \]
   fabs-sqr [=>] 48.2	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{-\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}\right)} \]
   sqr-pow [<=] 99.3	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{-\frac{\color{blue}{{x}^{1}}}{s}}\right)} \]
   unpow1 [=>] 99.3	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{-\frac{\color{blue}{x}}{s}}\right)} \]
   distribute-frac-neg [<=] 99.3	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\color{blue}{\frac{-x}{s}}}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{0 + \frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]
   Step-by-step derivation

   [Start] 99.3	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)} \]
   add-log-exp [=>] 79.2	\[ \color{blue}{\log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}}\right)} \]
   *-un-lft-identity [=>] 79.2	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}}\right)} \]
   log-prod [=>] 79.2	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}}\right)} \]
   metadata-eval [=>] 79.2	\[ \color{blue}{0} + \log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}}\right) \]
   add-log-exp [<=] 99.3	\[ 0 + \color{blue}{\frac{\frac{1}{s}}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}} \]
   associate-/l/ [=>] 99.7	\[ 0 + \color{blue}{\frac{1}{\left(2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)\right) \cdot s}} \]
   *-commutative [=>] 99.7	\[ 0 + \frac{1}{\color{blue}{s \cdot \left(2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)\right)}} \]
   +-commutative [=>] 99.7	\[ 0 + \frac{1}{s \cdot \color{blue}{\left(\left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right) + 2\right)}} \]
   distribute-frac-neg [=>] 99.7	\[ 0 + \frac{1}{s \cdot \left(\left(e^{\frac{x}{s}} + e^{\color{blue}{-\frac{x}{s}}}\right) + 2\right)} \]
   cosh-undef [=>] 99.7	\[ 0 + \frac{1}{s \cdot \left(\color{blue}{2 \cdot \cosh \left(\frac{x}{s}\right)} + 2\right)} \]
   fma-def [=>] 99.7	\[ 0 + \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]

8. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]
   Step-by-step derivation

   [Start] 99.7	\[ 0 + \frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)} \]
   +-lft-identity [=>] 99.7	\[ \color{blue}{\frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(2, \color{blue}{\frac{1}{\frac{2}{\cosh \left(\frac{x}{s}\right) \cdot 2}}}, 2\right)} \]
   Step-by-step derivation

   [Start] 99.7	\[ \frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)} \]
   cosh-def [=>] 99.7	\[ \frac{1}{s \cdot \mathsf{fma}\left(2, \color{blue}{\frac{e^{\frac{x}{s}} + e^{-\frac{x}{s}}}{2}}, 2\right)} \]
   clear-num [=>] 99.7	\[ \frac{1}{s \cdot \mathsf{fma}\left(2, \color{blue}{\frac{1}{\frac{2}{e^{\frac{x}{s}} + e^{-\frac{x}{s}}}}}, 2\right)} \]
   cosh-undef [=>] 99.7	\[ \frac{1}{s \cdot \mathsf{fma}\left(2, \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh \left(\frac{x}{s}\right)}}}, 2\right)} \]
   *-commutative [=>] 99.7	\[ \frac{1}{s \cdot \mathsf{fma}\left(2, \frac{1}{\frac{2}{\color{blue}{\cosh \left(\frac{x}{s}\right) \cdot 2}}}, 2\right)} \]

10. Final simplification 99.7%

    \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(2, \frac{1}{\frac{2}{2 \cdot \cosh \left(\frac{x}{s}\right)}}, 2\right)} \]

Alternatives

Alternative 1
Accuracy	96.3%
Cost	3588

\[\begin{array}{l} \mathbf{if}\;x \leq 4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + e^{\frac{-x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 2
Accuracy	95.4%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 4.000000094968912 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + t_0\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	3552

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

Alternative 4
Accuracy	91.4%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + \left(2 + \frac{x \cdot x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 5
Accuracy	60.7%
Cost	752

\[\begin{array}{l} t_0 := \frac{\frac{1}{s}}{2 + \left(2 + \frac{x \cdot x}{s \cdot s}\right)}\\ t_1 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq 5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	60.7%
Cost	688

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ t_1 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq 5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	60.7%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq 1.8000000068241206 \cdot 10^{-7} \lor \neg \left(x \leq 1.000000013351432 \cdot 10^{-10}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \end{array} \]

Alternative 8
Accuracy	60.7%
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq 9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 2.9999998795923744 \cdot 10^{-13}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \end{array} \]

Alternative 9
Accuracy	60.7%
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq 9.999999717180685 \cdot 10^{-10} \lor \neg \left(x \leq 2.9999998795923744 \cdot 10^{-13}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 10
Accuracy	43.2%
Cost	361

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6} \lor \neg \left(x \leq 1.9999999949504854 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11
Accuracy	43.0%
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12
Accuracy	41.7%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6} \lor \neg \left(x \leq 1.9999999949504854 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Accuracy	41.9%
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 14
Accuracy	27.9%
Cost	96

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

Reproduce

herbie shell --seed 2023159 
(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))))))