Average Error: 0.1 → 0.1
Time: 10.9s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}\]
↓
\[\frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + \left(2 + e^{-\frac{\left|x\right|}{s}}\right)\right)}\]
\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{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + \left(2 + e^{-\frac{\left|x\right|}{s}}\right)\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
 (/ 1.0 (* s (+ (exp (/ (fabs x) s)) (+ 2.0 (exp (- (/ (fabs x) 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 / (s * (expf(fabsf(x) / s) + (2.0f + expf(-(fabsf(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

    \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}\]
  2. Using strategy rm

  3. Applied add-log-exp_binary320.2

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

  4. Applied add-log-exp_binary320.3

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

  5. Applied sum-log_binary320.1

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

  6. Simplified0.1

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

  8. Applied associate-/r*_binary320.1

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

  9. Simplified0.2

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

  10. Taylor expanded around 0 0.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2021174 
(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))))))