Logistic regression 2

Average Error: 0.6 → 0.6

Time: 10.4s

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 r123087 = 1.0;
        double r123088 = x;
        double r123089 = exp(r123088);
        double r123090 = r123087 + r123089;
        double r123091 = log(r123090);
        double r123092 = y;
        double r123093 = r123088 * r123092;
        double r123094 = r123091 - r123093;
        return r123094;
}

↓

double f(double x, double y) {
        double r123095 = 1.0;
        double r123096 = x;
        double r123097 = exp(r123096);
        double r123098 = r123095 + r123097;
        double r123099 = log(r123098);
        double r123100 = y;
        double r123101 = r123096 * r123100;
        double r123102 = r123099 - r123101;
        return r123102;
}

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 
(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)))