Average Error: 0.5 → 1.0
Time: 12.9s
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 r7121817 = 1.0;
        double r7121818 = x;
        double r7121819 = exp(r7121818);
        double r7121820 = r7121817 + r7121819;
        double r7121821 = log(r7121820);
        double r7121822 = y;
        double r7121823 = r7121818 * r7121822;
        double r7121824 = r7121821 - r7121823;
        return r7121824;
}
↓
double f(double x, double y) {
        double r7121825 = 1.0;
        double r7121826 = x;
        double r7121827 = exp(r7121826);
        double r7121828 = r7121825 + r7121827;
        double r7121829 = sqrt(r7121828);
        double r7121830 = log(r7121829);
        double r7121831 = y;
        double r7121832 = r7121831 * r7121826;
        double r7121833 = r7121830 - r7121832;
        double r7121834 = r7121830 + r7121833;
        return r7121834;
}

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

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

Reproduce

herbie shell --seed 2019192 
(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)))