Average Error: 0.5 → 1.0
Time: 18.4s
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 r5133647 = 1.0;
        double r5133648 = x;
        double r5133649 = exp(r5133648);
        double r5133650 = r5133647 + r5133649;
        double r5133651 = log(r5133650);
        double r5133652 = y;
        double r5133653 = r5133648 * r5133652;
        double r5133654 = r5133651 - r5133653;
        return r5133654;
}
↓
double f(double x, double y) {
        double r5133655 = 1.0;
        double r5133656 = x;
        double r5133657 = exp(r5133656);
        double r5133658 = r5133655 + r5133657;
        double r5133659 = sqrt(r5133658);
        double r5133660 = log(r5133659);
        double r5133661 = y;
        double r5133662 = r5133661 * r5133656;
        double r5133663 = r5133660 - r5133662;
        double r5133664 = r5133660 + r5133663;
        return r5133664;
}

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
Target0.1
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 2019144 
(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)))