Logistic regression 2

Average Error: 0.6 → 1.1

Time: 4.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.650088164488972e+220
  2. y = 4.289362951266288e+268

Math

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

↓

\[\log \left(\sqrt{1 + e^{x}}\right) + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{1 + e^{x}}\right)\right)\right) - x \cdot y\right)\]

C

double code(double x, double y) {
	return (log((1.0 + exp(x))) - (x * y));
}

↓

double code(double x, double y) {
	return (log(sqrt((1.0 + exp(x)))) + (expm1(log1p(log(sqrt((1.0 + exp(x)))))) - (x * y)));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.6
Target	0.1
Herbie	1.1

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

Derivation

1. Initial program 0.6

   \[\log \left(1 + e^{x}\right) - x \cdot y\]
2. Using strategy rm

3. Applied add-sqr-sqrt 1.4

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

4. Applied log-prod 1.1

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

5. Applied associate--l+ 1.1

   \[\leadsto \color{blue}{\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\right)}\]
6. Using strategy rm

7. Applied expm1-log1p-u 1.1

   \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{1 + e^{x}}\right)\right)\right)} - x \cdot y\right)\]

8. Final simplification 1.1

   \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\sqrt{1 + e^{x}}\right)\right)\right) - x \cdot y\right)\]

Reproduce

herbie shell --seed 2020103 +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)))