Logistic regression 2

Average Error: 0.5 → 0.5

Time: 4.3s

Precision: binary64

Math

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

↓

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

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) {
	return ((double) (((double) exp(((double) cbrt(((double) pow(((double) log(((double) log(((double) (1.0 + ((double) exp(x)))))))), 3.0)))))) - ((double) (x * y))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 0.5

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

3. Applied add-exp-log 0.5

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

5. Applied add-cbrt-cube 0.5

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

6. Simplified 0.5

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

7. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020147 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (neg x)))) (* (neg x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))