Average Error: 0.2 → 0.2
Time: 6.8s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{s}}{{e}^{\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{-\left|x\right|}{s}} + 2\right)} \]
\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)}
↓
\frac{\frac{1}{s}}{{e}^{\left(\frac{\left|x\right|}{s}\right)} + \left(e^{\frac{-\left|x\right|}{s}} + 2\right)}

(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
 (/ (/ 1.0 s) (+ (pow E (/ (fabs x) s)) (+ (exp (/ (- (fabs x)) s)) 2.0))))
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) {
	return (1.0f / s) / (powf(((float) M_E), (fabsf(x) / s)) + (expf(-fabsf(x) / s) + 2.0f));
}

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 0.2

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

  2. 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. Applied *-un-lft-identity_binary320.2

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

  4. Applied *-un-lft-identity_binary320.2

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

  5. Applied fabs-mul_binary320.2

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

  6. Applied times-frac_binary320.2

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

  7. Applied exp-prod_binary320.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2022081 
(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))))))