Average Error: 0.1 → 0.0
Time: 4.5s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[{e}^{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)} \]
\frac{1}{1 + e^{\frac{-x}{s}}}
↓
{e}^{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)}

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s) :precision binary32 (pow E (- (log1p (exp (- (/ x s)))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf(-x / s));
}
↓
float code(float x, float s) {
	return powf(((float) M_E), -log1pf(expf(-(x / 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 0.1

    \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

  2. Applied add-exp-log_binary320.1

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

  3. Applied 1-exp_binary320.1

    \[\leadsto \frac{\color{blue}{e^{0}}}{e^{\log \left(1 + e^{\frac{-x}{s}}\right)}} \]

  4. Applied div-exp_binary320.1

    \[\leadsto \color{blue}{e^{0 - \log \left(1 + e^{\frac{-x}{s}}\right)}} \]

  5. Simplified0.0

    \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)}} \]

  6. Applied *-un-lft-identity_binary320.0

    \[\leadsto e^{\color{blue}{1 \cdot \left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)}} \]

  7. Applied exp-prod_binary320.0

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)}} \]

  8. Simplified0.0

    \[\leadsto {\color{blue}{e}}^{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)} \]

  9. Final simplification0.0

    \[\leadsto {e}^{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)} \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))