Average Error: 0.1 → 0.1
Time: 5.2s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[e^{-\mathsf{log1p}\left(\frac{1}{e^{\frac{x}{s}}}\right)} \]
\frac{1}{1 + e^{\frac{-x}{s}}}
↓
e^{-\mathsf{log1p}\left(\frac{1}{e^{\frac{x}{s}}}\right)}

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s) :precision binary32 (exp (- (log1p (/ 1.0 (exp (/ x s)))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf(-x / s));
}
↓
float code(float x, float s) {
	return expf(-log1pf(1.0f / 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. Simplified0.1

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

  6. Applied neg-sub0_binary320.1

    \[\leadsto e^{-\mathsf{log1p}\left(e^{\color{blue}{0 - \frac{x}{s}}}\right)} \]

  7. Applied exp-diff_binary320.1

    \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{\frac{e^{0}}{e^{\frac{x}{s}}}}\right)} \]

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021313 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))