Average Error: 0.5 → 0.5
Time: 15.9s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\mathsf{fma}\left(-y, x, \log \left(1 + e^{x}\right)\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\mathsf{fma}\left(-y, x, \log \left(1 + e^{x}\right)\right)
double f(double x, double y) {
        double r124804 = 1.0;
        double r124805 = x;
        double r124806 = exp(r124805);
        double r124807 = r124804 + r124806;
        double r124808 = log(r124807);
        double r124809 = y;
        double r124810 = r124805 * r124809;
        double r124811 = r124808 - r124810;
        return r124811;
}
↓
double f(double x, double y) {
        double r124812 = y;
        double r124813 = -r124812;
        double r124814 = x;
        double r124815 = 1.0;
        double r124816 = exp(r124814);
        double r124817 = r124815 + r124816;
        double r124818 = log(r124817);
        double r124819 = fma(r124813, r124814, r124818);
        return r124819;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.0
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, \log \left(1 + e^{x}\right)\right)}\]

  3. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(-y, x, \log \left(1 + e^{x}\right)\right)\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))