Logistic regression 2

Average Error: 0.4 → 0.4

Time: 6.2s

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 r157134 = 1.0;
        double r157135 = x;
        double r157136 = exp(r157135);
        double r157137 = r157134 + r157136;
        double r157138 = log(r157137);
        double r157139 = y;
        double r157140 = r157135 * r157139;
        double r157141 = r157138 - r157140;
        return r157141;
}

↓

double f(double x, double y) {
        double r157142 = 1.0;
        double r157143 = x;
        double r157144 = exp(r157143);
        double r157145 = r157142 + r157144;
        double r157146 = log(r157145);
        double r157147 = y;
        double r157148 = r157143 * r157147;
        double r157149 = r157146 - r157148;
        return r157149;
}

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)))