Logistic regression 2

Average Error: 0.5 → 0.6

Time: 4.7s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.864806516844435e+224
  2. y = 2.278825339876756e-210

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -915233.332766449894:\\ \;\;\;\;\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log 2 + \left({x}^{2} \cdot \left(\sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}} \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right)\right) \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right) + 0.5 \cdot x\right) - x \cdot y\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if ((x <= -915233.3327664499)) {
		VAR = ((double) (((double) log(((double) (((double) cbrt(((double) (1.0 + ((double) exp(x)))))) * ((double) cbrt(((double) (1.0 + ((double) exp(x)))))))))) + ((double) (((double) log(((double) cbrt(((double) (1.0 + ((double) exp(x)))))))) - ((double) (x * y))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) log(2.0)) + ((double) (((double) (((double) pow(x, 2.0)) * ((double) (((double) cbrt(((double) (0.25 - ((double) (0.5 / ((double) pow(2.0, 2.0)))))))) * ((double) cbrt(((double) (0.25 - ((double) (0.5 / ((double) pow(2.0, 2.0)))))))))))) * ((double) cbrt(((double) (0.25 - ((double) (0.5 / ((double) pow(2.0, 2.0)))))))))))) + ((double) (0.5 * x)))) - ((double) (x * y))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -915233.3327664499

   1. Initial program 0

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

   3. Applied add-cube-cbrt 0

      \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) \cdot \sqrt[3]{1 + e^{x}}\right)} - x \cdot y\]

   4. Applied log-prod 0

      \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \log \left(\sqrt[3]{1 + e^{x}}\right)\right)} - x \cdot y\]

   5. Applied associate--l+ 0

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)}\]

   if -915233.3327664499 < x

   1. Initial program 0.8

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

   2. Taylor expanded around 0 0.8

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

   3. Simplified 0.8

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

   5. Applied add-cube-cbrt 0.8

      \[\leadsto \left(\left(\log 2 + {x}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}} \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right) \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right)}\right) + 0.5 \cdot x\right) - x \cdot y\]

   6. Applied associate-*r* 0.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -915233.332766449894:\\ \;\;\;\;\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log 2 + \left({x}^{2} \cdot \left(\sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}} \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right)\right) \cdot \sqrt[3]{0.25 - \frac{\frac{1}{2}}{{2}^{2}}}\right) + 0.5 \cdot x\right) - x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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)))