Average Error: 0.1 → 0.1
Time: 7.8s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}\]
↓
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot 2 + \left({\left(e^{\frac{-\left|x\right|}{s}}\right)}^{2} + 1\right)\right)}\]
\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{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} \cdot 2 + \left({\left(e^{\frac{-\left|x\right|}{s}}\right)}^{2} + 1\right)\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
 (/
  (exp (/ (- (fabs x)) s))
  (*
   s
   (+
    (* (exp (/ (- (fabs x)) s)) 2.0)
    (+ (pow (exp (/ (- (fabs x)) s)) 2.0) 1.0)))))
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 expf(-fabsf(x) / s) / (s * ((expf(-fabsf(x) / s) * 2.0f) + (powf(expf(-fabsf(x) / s), 2.0f) + 1.0f)));
}

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 0.1

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

  2. Taylor expanded around 0 0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2021175 
(FPCore (x s)
  :name "Logistic"
  :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))))))