Logistic regression 2

Average Error: 0.6 → 1.1

Time: 3.1s

Precision: binary64

Math

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

↓

\[\sqrt[3]{e^{3 \cdot \log \log \left(1 + e^{x}\right)}} - x \cdot y\]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

↓

(FPCore (x y)
 :precision binary64
 (- (cbrt (exp (* 3.0 (log (log (+ 1.0 (exp x))))))) (* x y)))

C

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

↓

double code(double x, double y) {
	return cbrt(exp(3.0 * log(log(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

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

1. Initial program 0.6

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

3. Applied add-cbrt-cube_binary64_2132 0.6

   \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)\right) \cdot \log \left(1 + e^{x}\right)}} - x \cdot y\]

4. Simplified 0.6

   \[\leadsto \sqrt[3]{\color{blue}{{\log \left(1 + e^{x}\right)}^{3}}} - x \cdot y\]
5. Using strategy rm

6. Applied pow-to-exp_binary64_2165 1.1

   \[\leadsto \sqrt[3]{\color{blue}{e^{\log \log \left(1 + e^{x}\right) \cdot 3}}} - x \cdot y\]

7. Simplified 1.1

   \[\leadsto \sqrt[3]{e^{\color{blue}{3 \cdot \log \log \left(1 + e^{x}\right)}}} - x \cdot y\]

8. Final simplification 1.1

   \[\leadsto \sqrt[3]{e^{3 \cdot \log \log \left(1 + e^{x}\right)}} - x \cdot y\]

Reproduce

herbie shell --seed 2020280 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))