Average Error: 0.2 → 0.2
Time: 5.8s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{-\left|x\right|}{s}} + 2\right)}}{s} \]
\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}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{-\left|x\right|}{s}} + 2\right)}}{s}

(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 (+ (exp (/ (fabs x) s)) (+ (exp (/ (- (fabs x)) s)) 2.0))) 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) {
	return (1.0f / (expf(fabsf(x) / s) + (expf(-fabsf(x) / s) + 2.0f))) / s;
}

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 associate-/l/_binary320.2

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

  4. Applied associate-/r*_binary320.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2021313 
(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))))))