Average Error: 0.5 → 1.0
Time: 30.4s
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 r5902748 = 1.0;
        double r5902749 = x;
        double r5902750 = exp(r5902749);
        double r5902751 = r5902748 + r5902750;
        double r5902752 = log(r5902751);
        double r5902753 = y;
        double r5902754 = r5902749 * r5902753;
        double r5902755 = r5902752 - r5902754;
        return r5902755;
}
↓
double f(double x, double y) {
        double r5902756 = x;
        double r5902757 = exp(r5902756);
        double r5902758 = 1.0;
        double r5902759 = r5902757 + r5902758;
        double r5902760 = sqrt(r5902759);
        double r5902761 = log(r5902760);
        double r5902762 = y;
        double r5902763 = r5902762 * r5902756;
        double r5902764 = r5902761 - r5902763;
        double r5902765 = r5902764 + r5902761;
        return r5902765;
}

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)))