Average Error: 0.1 → 0.1
Time: 6.4s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{\left|x\right|}{s}}\\ \frac{1}{2 \cdot s + \left(\frac{s}{t_0} + s \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)} \end{array} \]
\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{1}{2 \cdot s + \left(\frac{s}{t_0} + s \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\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))))
   (/ 1.0 (+ (* 2.0 s) (+ (/ s t_0) (* s (expm1 (log1p 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 = expf(fabsf(x) / s);
	return 1.0f / ((2.0f * s) + ((s / t_0) + (s * expm1f(log1pf(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 0.1

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

  2. Simplified0.2

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

  3. Taylor expanded in s around 0 0.1

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

  4. Simplified0.1

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

  5. Taylor expanded in x around 0 0.1

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

  6. Applied expm1-log1p-u_binary320.1

    \[\leadsto \frac{1}{2 \cdot s + \left(\frac{s}{e^{\frac{\left|x\right|}{s}}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{s}}\right)\right)} \cdot s\right)} \]

  7. Final simplification0.1

    \[\leadsto \frac{1}{2 \cdot s + \left(\frac{s}{e^{\frac{\left|x\right|}{s}}} + s \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{s}}\right)\right)\right)} \]

Reproduce

herbie shell --seed 2021280 
(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))))))