Average Error: 0.5 → 1.0
Time: 16.3s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - y \cdot x\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - y \cdot x\right)
double f(double x, double y) {
        double r6718439 = 1.0;
        double r6718440 = x;
        double r6718441 = exp(r6718440);
        double r6718442 = r6718439 + r6718441;
        double r6718443 = log(r6718442);
        double r6718444 = y;
        double r6718445 = r6718440 * r6718444;
        double r6718446 = r6718443 - r6718445;
        return r6718446;
}
↓
double f(double x, double y) {
        double r6718447 = 1.0;
        double r6718448 = x;
        double r6718449 = exp(r6718448);
        double r6718450 = r6718447 + r6718449;
        double r6718451 = sqrt(r6718450);
        double r6718452 = log(r6718451);
        double r6718453 = y;
        double r6718454 = r6718453 * r6718448;
        double r6718455 = r6718452 - r6718454;
        double r6718456 = r6718452 + r6718455;
        return r6718456;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied add-sqr-sqrt1.3

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

  4. Applied log-prod1.0

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

  5. Applied associate--l+1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2019163 
(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)))