Average Error: 0.5 → 0.4
Time: 9.2s
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 r5741059 = 1.0;
        double r5741060 = x;
        double r5741061 = exp(r5741060);
        double r5741062 = r5741059 + r5741061;
        double r5741063 = log(r5741062);
        double r5741064 = y;
        double r5741065 = r5741060 * r5741064;
        double r5741066 = r5741063 - r5741065;
        return r5741066;
}
↓
double f(double x, double y) {
        double r5741067 = x;
        double r5741068 = exp(r5741067);
        double r5741069 = log1p(r5741068);
        double r5741070 = y;
        double r5741071 = r5741070 * r5741067;
        double r5741072 = r5741069 - r5741071;
        return r5741072;
}

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

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +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)))