Logistic regression 2

Average Error: 0.5 → 0.6

Time: 13.9s

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 r3058473 = 1.0;
        double r3058474 = x;
        double r3058475 = exp(r3058474);
        double r3058476 = r3058473 + r3058475;
        double r3058477 = log(r3058476);
        double r3058478 = y;
        double r3058479 = r3058474 * r3058478;
        double r3058480 = r3058477 - r3058479;
        return r3058480;
}

↓

double f(double x, double y) {
        double r3058481 = 1.0;
        double r3058482 = x;
        double r3058483 = exp(r3058482);
        double r3058484 = r3058483 * r3058483;
        double r3058485 = r3058483 * r3058484;
        double r3058486 = 3.0;
        double r3058487 = pow(r3058485, r3058486);
        double r3058488 = r3058481 + r3058487;
        double r3058489 = log(r3058488);
        double r3058490 = r3058484 * r3058484;
        double r3058491 = r3058490 - r3058483;
        double r3058492 = r3058491 * r3058484;
        double r3058493 = r3058481 + r3058492;
        double r3058494 = log(r3058493);
        double r3058495 = r3058489 - r3058494;
        double r3058496 = r3058483 - r3058481;
        double r3058497 = r3058484 - r3058496;
        double r3058498 = log(r3058497);
        double r3058499 = y;
        double r3058500 = r3058482 * r3058499;
        double r3058501 = r3058498 + r3058500;
        double r3058502 = r3058495 - r3058501;
        return r3058502;
}

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)))