Average Error: 0.5 → 0.5
Time: 15.6s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left({\left(e^{x}\right)}^{3} + 1\right) - \left(\log \left(\left(e^{x} \cdot e^{x} - e^{x}\right) - -1\right) + y \cdot x\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left({\left(e^{x}\right)}^{3} + 1\right) - \left(\log \left(\left(e^{x} \cdot e^{x} - e^{x}\right) - -1\right) + y \cdot x\right)
double f(double x, double y) {
        double r4527826 = 1.0;
        double r4527827 = x;
        double r4527828 = exp(r4527827);
        double r4527829 = r4527826 + r4527828;
        double r4527830 = log(r4527829);
        double r4527831 = y;
        double r4527832 = r4527827 * r4527831;
        double r4527833 = r4527830 - r4527832;
        return r4527833;
}
↓
double f(double x, double y) {
        double r4527834 = x;
        double r4527835 = exp(r4527834);
        double r4527836 = 3.0;
        double r4527837 = pow(r4527835, r4527836);
        double r4527838 = 1.0;
        double r4527839 = r4527837 + r4527838;
        double r4527840 = log(r4527839);
        double r4527841 = r4527835 * r4527835;
        double r4527842 = r4527841 - r4527835;
        double r4527843 = -1.0;
        double r4527844 = r4527842 - r4527843;
        double r4527845 = log(r4527844);
        double r4527846 = y;
        double r4527847 = r4527846 * r4527834;
        double r4527848 = r4527845 + r4527847;
        double r4527849 = r4527840 - r4527848;
        return r4527849;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+0.5

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

  4. Applied log-div0.5

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

  5. Applied associate--l-0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 
(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)))