Logistic regression 2

Average Error: 0.4 → 0.4

Time: 6.1s

Precision: 64

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

• could not determine a ground truth for program body

  1. x = 2.3826293734222903e+48
  2. y = -1.352832018891749e-22

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 r125155 = 1.0;
        double r125156 = x;
        double r125157 = exp(r125156);
        double r125158 = r125155 + r125157;
        double r125159 = log(r125158);
        double r125160 = y;
        double r125161 = r125156 * r125160;
        double r125162 = r125159 - r125161;
        return r125162;
}

↓

double f(double x, double y) {
        double r125163 = 1.0;
        double r125164 = x;
        double r125165 = exp(r125164);
        double r125166 = r125163 + r125165;
        double r125167 = log(r125166);
        double r125168 = y;
        double r125169 = r125164 * r125168;
        double r125170 = r125167 - r125169;
        return r125170;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.1
Herbie	0.4

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

Derivation

1. Initial program 0.4

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

2. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020042 
(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)))