Logistic regression 2

Average Error: 0.5 → 0.6

Time: 12.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.102410604960788e+155
  2. y = -1.508185319704459e+86

Math

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

↓

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

C

double f(double x, double y) {
        double r2876373 = 1.0;
        double r2876374 = x;
        double r2876375 = exp(r2876374);
        double r2876376 = r2876373 + r2876375;
        double r2876377 = log(r2876376);
        double r2876378 = y;
        double r2876379 = r2876374 * r2876378;
        double r2876380 = r2876377 - r2876379;
        return r2876380;
}

↓

double f(double x, double y) {
        double r2876381 = 1.0;
        double r2876382 = x;
        double r2876383 = exp(r2876382);
        double r2876384 = r2876383 * r2876383;
        double r2876385 = r2876383 * r2876384;
        double r2876386 = 3.0;
        double r2876387 = pow(r2876385, r2876386);
        double r2876388 = r2876381 + r2876387;
        double r2876389 = log(r2876388);
        double r2876390 = r2876384 * r2876384;
        double r2876391 = r2876390 - r2876383;
        double r2876392 = r2876391 * r2876384;
        double r2876393 = r2876381 + r2876392;
        double r2876394 = log(r2876393);
        double r2876395 = r2876389 - r2876394;
        double r2876396 = r2876383 - r2876381;
        double r2876397 = r2876384 - r2876396;
        double r2876398 = log(r2876397);
        double r2876399 = y;
        double r2876400 = r2876382 * r2876399;
        double r2876401 = r2876398 + r2876400;
        double r2876402 = r2876395 - r2876401;
        return r2876402;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.0
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

1. Initial program 0.5

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

3. Applied 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. Applied associate--l- 0.6

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

6. Simplified 0.6

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

8. Applied flip3-+ 0.6

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

9. Applied log-div 0.6

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

10. Simplified 0.6

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

11. Simplified 0.6

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

12. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

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