Average Error: 0.2 → 0.1
Time: 9.4s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{{\left(\sqrt{e^{-1}}\right)}^{\left(t_0 \cdot 2\right)}}{\mathsf{fma}\left(s, e^{t_0 \cdot -2} + \frac{2}{e^{t_0}}, s\right)} \end{array} \]
\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{{\left(\sqrt{e^{-1}}\right)}^{\left(t_0 \cdot 2\right)}}{\mathsf{fma}\left(s, e^{t_0 \cdot -2} + \frac{2}{e^{t_0}}, s\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 (sqrt (exp -1.0)) (* t_0 2.0))
    (fma s (+ (exp (* t_0 -2.0)) (/ 2.0 (exp t_0))) s))))
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(sqrtf(expf(-1.0f)), (t_0 * 2.0f)) / fmaf(s, (expf(t_0 * -2.0f) + (2.0f / expf(t_0))), s);
}

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. 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 neg-mul-1_binary320.2

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

  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

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

  7. Applied add-sqr-sqrt_binary320.2

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

  8. Applied unpow-prod-down_binary320.2

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

  9. Applied times-frac_binary320.1

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

  10. Simplified0.1

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

  11. Simplified0.1

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

  12. Taylor expanded in s around 0 0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(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))))))