Logistic regression 2

Average Error: 0.6 → 0.6

Time: 9.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 10920358539895.18
  2. y = 9.278264258908537e-20

Math

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

↓

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

C

double f(double x, double y) {
        double r150796 = 1.0;
        double r150797 = x;
        double r150798 = exp(r150797);
        double r150799 = r150796 + r150798;
        double r150800 = log(r150799);
        double r150801 = y;
        double r150802 = r150797 * r150801;
        double r150803 = r150800 - r150802;
        return r150803;
}

↓

double f(double x, double y) {
        double r150804 = 1.0;
        double r150805 = x;
        double r150806 = exp(r150805);
        double r150807 = r150804 + r150806;
        double r150808 = log(r150807);
        double r150809 = y;
        double r150810 = r150805 * r150809;
        double r150811 = r150808 - r150810;
        return r150811;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.6
Target	0.1
Herbie	0.6

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

Derivation

1. Initial program 0.6

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

2. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020046 
(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)))