Average Error: 0.2 → 0.1
Time: 7.6s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := -\frac{\left|x\right|}{s}\\ t_1 := {e}^{\left(\frac{t_0}{2}\right)}\\ \frac{1}{\frac{s}{t_1} \cdot \frac{{\left(1 + e^{t_0}\right)}^{2}}{t_1}} \end{array} \]
\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 := -\frac{\left|x\right|}{s}\\
t_1 := {e}^{\left(\frac{t_0}{2}\right)}\\
\frac{1}{\frac{s}{t_1} \cdot \frac{{\left(1 + e^{t_0}\right)}^{2}}{t_1}}
\end{array}

(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 (- (/ (fabs x) s))) (t_1 (pow E (/ t_0 2.0))))
   (/ 1.0 (* (/ s t_1) (/ (pow (+ 1.0 (exp t_0)) 2.0) t_1)))))
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 = -(fabsf(x) / s);
	float t_1 = powf(((float) M_E), (t_0 / 2.0f));
	return 1.0f / ((s / t_1) * (powf((1.0f + expf(t_0)), 2.0f) / t_1));
}

Error

Bits error versus x

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Applied clear-num_binary320.2

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

  3. Simplified0.1

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

  4. Applied *-un-lft-identity_binary320.1

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

  5. Applied exp-prod_binary320.2

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

  6. Simplified0.2

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

  7. Applied sqr-pow_binary320.2

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

  8. Applied times-frac_binary320.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022004 
(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))))))