Logistic regression 2

Average Error: 0.5 → 0.5

Time: 14.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.9540437202843566e+32
  2. y = -1.7431458063348651e-140

Math

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

↓

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

C

double f(double x, double y) {
        double r187655 = 1.0;
        double r187656 = x;
        double r187657 = exp(r187656);
        double r187658 = r187655 + r187657;
        double r187659 = log(r187658);
        double r187660 = y;
        double r187661 = r187656 * r187660;
        double r187662 = r187659 - r187661;
        return r187662;
}

↓

double f(double x, double y) {
        double r187663 = y;
        double r187664 = -r187663;
        double r187665 = x;
        double r187666 = 1.0;
        double r187667 = exp(r187665);
        double r187668 = r187666 + r187667;
        double r187669 = log(r187668);
        double r187670 = fma(r187664, r187665, r187669);
        return r187670;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.1
Herbie	0.5

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

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

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

3. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019195 +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)))