Logistic regression 2

Average Error: 0.6 → 0.6

Time: 12.6s

Precision: 64

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

• could not determine a ground truth for program body

  1. x = 8.474771072676942e+81
  2. y = -1.50113664730667e+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 r174138 = 1.0;
        double r174139 = x;
        double r174140 = exp(r174139);
        double r174141 = r174138 + r174140;
        double r174142 = log(r174141);
        double r174143 = y;
        double r174144 = r174139 * r174143;
        double r174145 = r174142 - r174144;
        return r174145;
}

↓

double f(double x, double y) {
        double r174146 = 1.0;
        double r174147 = x;
        double r174148 = exp(r174147);
        double r174149 = r174146 + r174148;
        double r174150 = log(r174149);
        double r174151 = y;
        double r174152 = r174147 * r174151;
        double r174153 = r174150 - r174152;
        return r174153;
}

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 2019350 +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)))