Average Error: 0.1 → 0.2
Time: 6.7s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{\mathsf{fma}\left(s, e^{-t_0}, s\right)} \cdot e^{-\mathsf{log1p}\left(e^{t_0}\right)} \end{array} \]
\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{1}{\mathsf{fma}\left(s, e^{-t_0}, s\right)} \cdot e^{-\mathsf{log1p}\left(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)))
   (* (/ 1.0 (fma s (exp (- t_0)) s)) (exp (- (log1p (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 (1.0f / fmaf(s, expf(-t_0), s)) * expf(-log1pf(expf(t_0)));
}

Error

Bits error versus x

Bits error versus s

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. Applied *-un-lft-identity_binary320.1

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

  3. Applied times-frac_binary320.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Applied add-exp-log_binary320.2

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

  7. Applied div-exp_binary320.2

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

  8. Simplified0.2

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)} \cdot e^{\color{blue}{-\mathsf{log1p}\left(e^{\frac{\left|x\right|}{s}}\right)}} \]

  9. Final simplification0.2

    \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)} \cdot e^{-\mathsf{log1p}\left(e^{\frac{\left|x\right|}{s}}\right)} \]

Reproduce

herbie shell --seed 2022125 
(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))))))