Average Error: 0.1 → 0.2
Time: 7.9s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{s} \cdot \frac{-1}{-2 - \left(e^{-t_0} + e^{t_0}\right)} \end{array} \]
\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}{s} \cdot \frac{-1}{-2 - \left(e^{-t_0} + 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 s) (/ -1.0 (- -2.0 (+ (exp (- t_0)) (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 / s) * (-1.0f / (-2.0f - (expf(-t_0) + expf(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. Using strategy rm

  4. Applied div-inv_binary320.2

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

  5. Simplified0.2

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

  7. Applied frac-2neg_binary320.2

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2021204 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (<= 0.0 s 1.0651631)
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))