Average Error: 0.5 → 0.4
Time: 18.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log_* (1 + e^{x}) - y \cdot x\]
double f(double x, double y) {
        double r5068122 = 1.0;
        double r5068123 = x;
        double r5068124 = exp(r5068123);
        double r5068125 = r5068122 + r5068124;
        double r5068126 = log(r5068125);
        double r5068127 = y;
        double r5068128 = r5068123 * r5068127;
        double r5068129 = r5068126 - r5068128;
        return r5068129;
}
↓
double f(double x, double y) {
        double r5068130 = x;
        double r5068131 = exp(r5068130);
        double r5068132 = log1p(r5068131);
        double r5068133 = y;
        double r5068134 = r5068133 * r5068130;
        double r5068135 = r5068132 - r5068134;
        return r5068135;
}


\log \left(1 + e^{x}\right) - x \cdot y
↓
\log_* (1 + e^{x}) - y \cdot x

Error

Bits error versus x

Bits error versus y

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

    \[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]

  3. Final simplification0.4

    \[\leadsto \log_* (1 + e^{x}) - y \cdot x\]

Reproduce

herbie shell --seed 2019101 +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)))