Logistic regression 2

Average Error: 0.4 → 0.4

Time: 6.3s

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 r173315 = 1.0;
        double r173316 = x;
        double r173317 = exp(r173316);
        double r173318 = r173315 + r173317;
        double r173319 = log(r173318);
        double r173320 = y;
        double r173321 = r173316 * r173320;
        double r173322 = r173319 - r173321;
        return r173322;
}

↓

double f(double x, double y) {
        double r173323 = 1.0;
        double r173324 = x;
        double r173325 = exp(r173324);
        double r173326 = r173323 + r173325;
        double r173327 = log(r173326);
        double r173328 = y;
        double r173329 = r173324 * r173328;
        double r173330 = r173327 - r173329;
        return r173330;
}

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 +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)))