Logistic regression 2

Average Error: 0.6 → 0.6

Time: 4.0s

Precision: 64

Math

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

↓

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

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

Error

Bits error versus x

Bits error versus y

Target

Original	0.6
Target	0.1
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. Initial program 0.6

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

3. Applied flip3-+ 0.6

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

4. Applied log-div 0.6

   \[\leadsto \color{blue}{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)} - x \cdot y\]
5. Using strategy rm

6. Applied add-cube-cbrt 0.6

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

8. Applied add-sqr-sqrt 0.6

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

9. Applied log-prod 0.6

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

11. Applied add-cbrt-cube 0.6

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

12. Simplified 0.6

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

13. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020130 
(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 (- x)))) (* (- x) (- 1.0 y))))

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