Average Error: 0.5 → 1.0
Time: 29.0s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{e^{x} + 1}\right) - y \cdot x\right) + \log \left(\sqrt{e^{x} + 1}\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{e^{x} + 1}\right) - y \cdot x\right) + \log \left(\sqrt{e^{x} + 1}\right)
double f(double x, double y) {
        double r5686685 = 1.0;
        double r5686686 = x;
        double r5686687 = exp(r5686686);
        double r5686688 = r5686685 + r5686687;
        double r5686689 = log(r5686688);
        double r5686690 = y;
        double r5686691 = r5686686 * r5686690;
        double r5686692 = r5686689 - r5686691;
        return r5686692;
}
↓
double f(double x, double y) {
        double r5686693 = x;
        double r5686694 = exp(r5686693);
        double r5686695 = 1.0;
        double r5686696 = r5686694 + r5686695;
        double r5686697 = sqrt(r5686696);
        double r5686698 = log(r5686697);
        double r5686699 = y;
        double r5686700 = r5686699 * r5686693;
        double r5686701 = r5686698 - r5686700;
        double r5686702 = r5686701 + r5686698;
        return r5686702;
}

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

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

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))