Logistic regression 2

Average Error: 0.5 → 0.6

Time: 4.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 7.873575565237604e+272
  2. y = 2.4935598987101598e+306

Math

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

↓

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

C

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

↓

double code(double x, double y) {
	return ((log((pow(pow(1.0, 3.0), 3.0) + pow(pow(exp(x), 3.0), 3.0))) - log(((pow(1.0, 3.0) * pow(1.0, 3.0)) + ((pow(exp(x), 3.0) * pow(exp(x), 3.0)) - (pow(1.0, 3.0) * pow(exp(x), 3.0)))))) - fma(y, x, log(((1.0 * 1.0) + ((exp(x) * exp(x)) - (1.0 * exp(x)))))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.0
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 0.5

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

3. Applied flip3-+ 0.6

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

4. Applied log-div 0.6

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

5. Applied associate--l- 0.6

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

6. Simplified 0.6

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

8. Applied flip3-+ 0.6

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

9. Applied log-div 0.6

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

10. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020071 +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)))