Average Error: 0.6 → 0.5
Time: 1.6m
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 r18983057 = 1.0;
        double r18983058 = x;
        double r18983059 = exp(r18983058);
        double r18983060 = r18983057 + r18983059;
        double r18983061 = log(r18983060);
        double r18983062 = y;
        double r18983063 = r18983058 * r18983062;
        double r18983064 = r18983061 - r18983063;
        return r18983064;
}
↓
double f(double x, double y) {
        double r18983065 = x;
        double r18983066 = exp(r18983065);
        double r18983067 = log1p(r18983066);
        double r18983068 = y;
        double r18983069 = r18983068 * r18983065;
        double r18983070 = r18983067 - r18983069;
        return r18983070;
}

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
Target0.0
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

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019120 +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)))