Logistic regression 2

Average Error: 0.5 → 0.7

Time: 5.6s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.479121160436394e+34
  2. y = -1.6040526032817678e-230

Math

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

↓

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

C

double f(double x, double y) {
        double r210417 = 1.0;
        double r210418 = x;
        double r210419 = exp(r210418);
        double r210420 = r210417 + r210419;
        double r210421 = log(r210420);
        double r210422 = y;
        double r210423 = r210418 * r210422;
        double r210424 = r210421 - r210423;
        return r210424;
}

↓

double f(double x, double y) {
        double r210425 = 1.0;
        double r210426 = 3.0;
        double r210427 = pow(r210425, r210426);
        double r210428 = x;
        double r210429 = exp(r210428);
        double r210430 = pow(r210429, r210426);
        double r210431 = r210427 + r210430;
        double r210432 = log(r210431);
        double r210433 = r210425 * r210425;
        double r210434 = r210429 * r210429;
        double r210435 = r210425 * r210429;
        double r210436 = r210434 - r210435;
        double r210437 = r210433 + r210436;
        double r210438 = log(r210437);
        double r210439 = cbrt(r210438);
        double r210440 = r210439 * r210439;
        double r210441 = exp(r210429);
        double r210442 = r210429 - r210425;
        double r210443 = pow(r210441, r210442);
        double r210444 = log(r210443);
        double r210445 = r210433 + r210444;
        double r210446 = log(r210445);
        double r210447 = cbrt(r210446);
        double r210448 = r210440 * r210447;
        double r210449 = y;
        double r210450 = r210428 * r210449;
        double r210451 = r210448 + r210450;
        double r210452 = r210432 - r210451;
        return r210452;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.1
Herbie	0.7

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

Derivation

1. Initial program 0.5

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

3. Applied flip3-+ 0.5

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

4. Applied log-div 0.5

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

5. Applied associate--l- 0.5

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

7. Applied add-cube-cbrt 0.5

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

9. Applied add-log-exp 0.6

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

10. Applied add-log-exp 0.7

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

11. Applied diff-log 0.7

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

12. Simplified 0.7

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

13. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020047 
(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)))