Logistic regression 2

Average Error: 0.5 → 0.4

Time: 4.2s

Precision: 64

Math

\[\log \left(1 + e^{x}\right) - x \cdot y\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.61400379247107798 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 \cdot 1 - e^{x} \cdot e^{x}}\right) - \log \left(\sqrt{1 - e^{x}}\right)\right)\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log 2 + {x}^{2} \cdot \left(0.25 - \frac{\frac{1}{2}}{{2}^{2}}\right)\right) + 0.5 \cdot x\right) - x \cdot y\\ \end{array}\]

C

double code(double x, double y) {
	return (log((1.0 + exp(x))) - (x * y));
}

↓

double code(double x, double y) {
	double VAR;
	if ((x <= -0.0008614003792471078)) {
		VAR = ((log(sqrt((1.0 + exp(x)))) + (log(sqrt(((1.0 * 1.0) - (exp(x) * exp(x))))) - log(sqrt((1.0 - exp(x)))))) - (x * y));
	} else {
		VAR = (((log(2.0) + (pow(x, 2.0) * (0.25 - (0.5 / pow(2.0, 2.0))))) + (0.5 * x)) - (x * y));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.1
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.0008614003792471078

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied log-prod 0.1

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right)} - x \cdot y\]
   5. Using strategy rm

   6. Applied flip-+ 0.2

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

   7. Applied sqrt-div 0.2

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

   8. Applied log-div 0.1

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

   if -0.0008614003792471078 < x

   1. Initial program 0.7

      \[\log \left(1 + e^{x}\right) - x \cdot y\]

   2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\left(\left(\log 2 + \left(0.25 \cdot {x}^{2} + 0.5 \cdot x\right)\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{2}^{2}}\right)} - x \cdot y\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\left(\left(\log 2 + {x}^{2} \cdot \left(0.25 - \frac{\frac{1}{2}}{{2}^{2}}\right)\right) + 0.5 \cdot x\right)} - x \cdot y\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.61400379247107798 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 \cdot 1 - e^{x} \cdot e^{x}}\right) - \log \left(\sqrt{1 - e^{x}}\right)\right)\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log 2 + {x}^{2} \cdot \left(0.25 - \frac{\frac{1}{2}}{{2}^{2}}\right)\right) + 0.5 \cdot x\right) - x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))