Average Error: 0.2 → 0.1
Time: 9.9s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{-t_0}\\ \frac{1}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(t_1, 2, {\left(e^{t_0}\right)}^{-2}\right), s\right)}{t_1}} \end{array} \]
\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^{-t_0}\\
\frac{1}{\frac{\mathsf{fma}\left(s, \mathsf{fma}\left(t_1, 2, {\left(e^{t_0}\right)}^{-2}\right), s\right)}{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 (exp (- t_0))))
   (/ 1.0 (/ (fma s (fma t_1 2.0 (pow (exp t_0) -2.0)) s) 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 = expf(-t_0);
	return 1.0f / (fmaf(s, fmaf(t_1, 2.0f, powf(expf(t_0), -2.0f)), s) / t_1);
}

Error

Bits error versus x

Bits error versus s

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

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

  3. Applied clear-num_binary320.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022082 
(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))))))