Average Error: 0.6 → 0.5
Time: 10.0s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\mathsf{log1p}\left(e^{x}\right) - y \cdot x\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\mathsf{log1p}\left(e^{x}\right) - y \cdot x
double f(double x, double y) {
        double r1907738 = 1.0;
        double r1907739 = x;
        double r1907740 = exp(r1907739);
        double r1907741 = r1907738 + r1907740;
        double r1907742 = log(r1907741);
        double r1907743 = y;
        double r1907744 = r1907739 * r1907743;
        double r1907745 = r1907742 - r1907744;
        return r1907745;
}
↓
double f(double x, double y) {
        double r1907746 = x;
        double r1907747 = exp(r1907746);
        double r1907748 = log1p(r1907747);
        double r1907749 = y;
        double r1907750 = r1907749 * r1907746;
        double r1907751 = r1907748 - r1907750;
        return r1907751;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.1
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 0.6

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - y \cdot x}\]

  3. Final simplification0.5

    \[\leadsto \mathsf{log1p}\left(e^{x}\right) - y \cdot x\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))