Average Error: 0.2 → 0.2
Time: 11.1s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}\]
↓
\[\frac{\frac{\sqrt{\frac{1}{e^{\frac{\left|x\right|}{s}} + \left(e^{-\frac{\left|x\right|}{s}} + 2\right)}}}{s}}{\sqrt{e^{\frac{\left|x\right|}{s}} + \left(e^{-\frac{\left|x\right|}{s}} + 2\right)}}\]
\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{\sqrt{\frac{1}{e^{\frac{\left|x\right|}{s}} + \left(e^{-\frac{\left|x\right|}{s}} + 2\right)}}}{s}}{\sqrt{e^{\frac{\left|x\right|}{s}} + \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
 (/
  (/
   (sqrt (/ 1.0 (+ (exp (/ (fabs x) s)) (+ (exp (- (/ (fabs x) s))) 2.0))))
   s)
  (sqrt (+ (exp (/ (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 (sqrtf(1.0f / (expf(fabsf(x) / s) + (expf(-(fabsf(x) / s)) + 2.0f))) / s) / sqrtf(expf(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. Using strategy rm

  4. Applied add-sqr-sqrt_binary320.2

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

  5. Applied associate-/r*_binary320.2

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

  6. Simplified0.2

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

  7. Taylor expanded around 0 0.2

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

  8. Simplified0.2

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

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

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2021173 
(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))))))