Average Error: 0.2 → 0.2
Time: 8.8s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ \frac{{e}^{t_0}}{\left(s \cdot \left(1 + \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(1 + e^{t_0}\right)} \end{array} \]
\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}\\
\frac{{e}^{t_0}}{\left(s \cdot \left(1 + \frac{1}{e^{\frac{\left|x\right|}{s}}}\right)\right) \cdot \left(1 + e^{t_0}\right)}
\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)))
   (/
    (pow E t_0)
    (* (* s (+ 1.0 (/ 1.0 (exp (/ (fabs x) s))))) (+ 1.0 (exp t_0))))))
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;
	return powf(((float) M_E), t_0) / ((s * (1.0f + (1.0f / expf(fabsf(x) / s)))) * (1.0f + expf(t_0)));
}

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 *-un-lft-identity_binary320.2

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

  3. Applied *-un-lft-identity_binary320.2

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

  4. Applied times-frac_binary320.2

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

  5. Applied exp-prod_binary320.2

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

  6. Simplified0.2

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

  7. Applied neg-sub0_binary320.2

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

  8. Applied div-sub_binary320.2

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

  9. Applied exp-diff_binary320.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2021215 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (<= 0.0 s 1.0651631)
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))