Average Error: 0.5 → 0.4
Time: 1.2m
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\mathsf{log1p}\left(\left(e^{x}\right)\right) - y \cdot x\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\mathsf{log1p}\left(\left(e^{x}\right)\right) - y \cdot x
double f(double x, double y) {
        double r11726883 = 1.0;
        double r11726884 = x;
        double r11726885 = exp(r11726884);
        double r11726886 = r11726883 + r11726885;
        double r11726887 = log(r11726886);
        double r11726888 = y;
        double r11726889 = r11726884 * r11726888;
        double r11726890 = r11726887 - r11726889;
        return r11726890;
}
↓
double f(double x, double y) {
        double r11726891 = x;
        double r11726892 = exp(r11726891);
        double r11726893 = log1p(r11726892);
        double r11726894 = y;
        double r11726895 = r11726894 * r11726891;
        double r11726896 = r11726893 - r11726895;
        return r11726896;
}

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019125 +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)))