Average Error: 0.5 → 0.5
Time: 16.5s
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 r89150 = 1.0;
        double r89151 = x;
        double r89152 = exp(r89151);
        double r89153 = r89150 + r89152;
        double r89154 = log(r89153);
        double r89155 = y;
        double r89156 = r89151 * r89155;
        double r89157 = r89154 - r89156;
        return r89157;
}
↓
double f(double x, double y) {
        double r89158 = y;
        double r89159 = x;
        double r89160 = -r89159;
        double r89161 = 1.0;
        double r89162 = exp(r89159);
        double r89163 = r89161 + r89162;
        double r89164 = log(r89163);
        double r89165 = fma(r89158, r89160, r89164);
        return r89165;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.1
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. Taylor expanded around inf 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019323 +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)))