Average Error: 0.2 → 0.1
Time: 8.2s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t_0}{s \cdot \left(1 + \left(t_0 \cdot 2 + {t_0}^{2}\right)\right)} \end{array} \]
\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 := e^{-\frac{\left|x\right|}{s}}\\
\frac{t_0}{s \cdot \left(1 + \left(t_0 \cdot 2 + {t_0}^{2}\right)\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 (exp (- (/ (fabs x) s)))))
   (/ t_0 (* s (+ 1.0 (+ (* t_0 2.0) (pow t_0 2.0)))))))
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 = expf(-(fabsf(x) / s));
	return t_0 / (s * (1.0f + ((t_0 * 2.0f) + powf(t_0, 2.0f))));
}

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. Taylor expanded in s around 0 0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022024 
(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))))))