Average Error: 0.5 → 1.0
Time: 14.2s
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 r7272676 = 1.0;
        double r7272677 = x;
        double r7272678 = exp(r7272677);
        double r7272679 = r7272676 + r7272678;
        double r7272680 = log(r7272679);
        double r7272681 = y;
        double r7272682 = r7272677 * r7272681;
        double r7272683 = r7272680 - r7272682;
        return r7272683;
}
↓
double f(double x, double y) {
        double r7272684 = 1.0;
        double r7272685 = x;
        double r7272686 = exp(r7272685);
        double r7272687 = r7272684 + r7272686;
        double r7272688 = sqrt(r7272687);
        double r7272689 = log(r7272688);
        double r7272690 = r7272689 + r7272689;
        double r7272691 = y;
        double r7272692 = r7272685 * r7272691;
        double r7272693 = r7272690 - r7272692;
        return r7272693;
}

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. 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 2019158 
(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)))