Average Error: 0.4 → 1.0
Time: 9.9s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y
double f(double x, double y) {
        double r1938914 = 1.0;
        double r1938915 = x;
        double r1938916 = exp(r1938915);
        double r1938917 = r1938914 + r1938916;
        double r1938918 = log(r1938917);
        double r1938919 = y;
        double r1938920 = r1938915 * r1938919;
        double r1938921 = r1938918 - r1938920;
        return r1938921;
}
↓
double f(double x, double y) {
        double r1938922 = 1.0;
        double r1938923 = x;
        double r1938924 = exp(r1938923);
        double r1938925 = r1938922 + r1938924;
        double r1938926 = sqrt(r1938925);
        double r1938927 = log(r1938926);
        double r1938928 = r1938927 + r1938927;
        double r1938929 = y;
        double r1938930 = r1938923 * r1938929;
        double r1938931 = r1938928 - r1938930;
        return r1938931;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
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 0.4

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  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. Final simplification1.0

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

Reproduce

herbie shell --seed 2019154 
(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)))