Average Error: 0.6 → 0.6
Time: 13.5s
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 r127584 = 1.0;
        double r127585 = x;
        double r127586 = exp(r127585);
        double r127587 = r127584 + r127586;
        double r127588 = log(r127587);
        double r127589 = y;
        double r127590 = r127585 * r127589;
        double r127591 = r127588 - r127590;
        return r127591;
}
↓
double f(double x, double y) {
        double r127592 = y;
        double r127593 = x;
        double r127594 = -r127593;
        double r127595 = 1.0;
        double r127596 = exp(r127593);
        double r127597 = r127595 + r127596;
        double r127598 = log(r127597);
        double r127599 = fma(r127592, r127594, r127598);
        return r127599;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.6
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around inf 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))