Logistic regression 2

Average Error: 0.4 → 1.0

Time: 16.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 17491141879405.176
  2. y = 2.330187764053037e+289

Math

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

↓

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

C

double f(double x, double y) {
        double r5189248 = 1.0;
        double r5189249 = x;
        double r5189250 = exp(r5189249);
        double r5189251 = r5189248 + r5189250;
        double r5189252 = log(r5189251);
        double r5189253 = y;
        double r5189254 = r5189249 * r5189253;
        double r5189255 = r5189252 - r5189254;
        return r5189255;
}

↓

double f(double x, double y) {
        double r5189256 = 1.0;
        double r5189257 = x;
        double r5189258 = exp(r5189257);
        double r5189259 = r5189256 + r5189258;
        double r5189260 = sqrt(r5189259);
        double r5189261 = log(r5189260);
        double r5189262 = r5189261 * r5189261;
        double r5189263 = r5189261 * r5189262;
        double r5189264 = cbrt(r5189263);
        double r5189265 = 3.0;
        double r5189266 = pow(r5189258, r5189265);
        double r5189267 = r5189266 + r5189256;
        double r5189268 = sqrt(r5189267);
        double r5189269 = log(r5189268);
        double r5189270 = r5189264 + r5189269;
        double r5189271 = -1.0;
        double r5189272 = r5189271 + r5189258;
        double r5189273 = r5189272 * r5189258;
        double r5189274 = r5189273 - r5189271;
        double r5189275 = sqrt(r5189274);
        double r5189276 = log(r5189275);
        double r5189277 = y;
        double r5189278 = r5189257 * r5189277;
        double r5189279 = r5189276 + r5189278;
        double r5189280 = r5189270 - r5189279;
        return r5189280;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.1
Herbie	1.0

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

Derivation

1. Initial program 0.4

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

3. Applied add-sqr-sqrt 1.2

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

4. Applied log-prod 0.9

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

6. Applied flip3-+ 1.0

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

7. Applied sqrt-div 1.0

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

8. Applied log-div 1.0

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

9. Applied associate-+r- 1.0

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

10. Applied associate--l- 1.0

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

11. Simplified 1.0

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

13. Applied add-cbrt-cube 1.0

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

14. Final simplification 1.0

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

Reproduce

herbie shell --seed 2019138 
(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)))