Average Error: 0.4 → 0.4
Time: 39.7s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log_* (1 + e^{x}) - y \cdot x\]
double f(double x, double y) {
        double r40277291 = 1.0;
        double r40277292 = x;
        double r40277293 = exp(r40277292);
        double r40277294 = r40277291 + r40277293;
        double r40277295 = log(r40277294);
        double r40277296 = y;
        double r40277297 = r40277292 * r40277296;
        double r40277298 = r40277295 - r40277297;
        return r40277298;
}
↓
double f(double x, double y) {
        double r40277299 = x;
        double r40277300 = exp(r40277299);
        double r40277301 = log1p(r40277300);
        double r40277302 = y;
        double r40277303 = r40277302 * r40277299;
        double r40277304 = r40277301 - r40277303;
        return r40277304;
}


\log \left(1 + e^{x}\right) - x \cdot y
↓
\log_* (1 + e^{x}) - y \cdot x

Error

Bits error versus x

Bits error versus y

Target

Original0.4
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 0.4

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]

  3. Final simplification0.4

    \[\leadsto \log_* (1 + e^{x}) - y \cdot x\]

Reproduce

herbie shell --seed 2019102 +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)))