Average Error: 0.5 → 0.4
Time: 13.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log_* (1 + e^{x}) - y \cdot x\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log_* (1 + e^{x}) - y \cdot x
double f(double x, double y) {
        double r17795345 = 1.0;
        double r17795346 = x;
        double r17795347 = exp(r17795346);
        double r17795348 = r17795345 + r17795347;
        double r17795349 = log(r17795348);
        double r17795350 = y;
        double r17795351 = r17795346 * r17795350;
        double r17795352 = r17795349 - r17795351;
        return r17795352;
}
↓
double f(double x, double y) {
        double r17795353 = x;
        double r17795354 = exp(r17795353);
        double r17795355 = log1p(r17795354);
        double r17795356 = y;
        double r17795357 = r17795356 * r17795353;
        double r17795358 = r17795355 - r17795357;
        return r17795358;
}

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

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 +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)))