Average Error: 0.1 → 0.1
Time: 3.9s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[\frac{1}{1 + {\left(\sqrt{e^{-1}}\right)}^{\left(2 \cdot \frac{x}{s}\right)}} \]
\frac{1}{1 + e^{\frac{-x}{s}}}
↓
\frac{1}{1 + {\left(\sqrt{e^{-1}}\right)}^{\left(2 \cdot \frac{x}{s}\right)}}

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (sqrt (exp -1.0)) (* 2.0 (/ x s))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf(-x / s));
}
↓
float code(float x, float s) {
	return 1.0f / (1.0f + powf(sqrtf(expf(-1.0f)), (2.0f * (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 *-un-lft-identity_binary320.1

    \[\leadsto \frac{1}{1 + e^{\frac{-x}{\color{blue}{1 \cdot s}}}} \]

  3. Applied neg-mul-1_binary320.1

    \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{-1 \cdot x}}{1 \cdot s}}} \]

  4. Applied times-frac_binary320.1

    \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{-1}{1} \cdot \frac{x}{s}}}} \]

  5. Applied exp-prod_binary320.1

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

  6. Simplified0.1

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

  7. Applied add-sqr-sqrt_binary320.1

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

  8. Applied unpow-prod-down_binary320.1

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

  9. Applied pow-prod-up_binary320.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))