Logistic regression 2

Average Error: 0.5 → 0.5

Time: 8.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.360512383203929e+29
  2. y = -1.0880729938714735e+237

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.1
Herbie	0.5

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

Derivation

1. Initial program 0.5

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

3. Applied add-cbrt-cube 0.5

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

4. Simplified 0.5

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

6. Applied add-sqr-sqrt 1.0

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

7. Applied pow-unpow 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 1.0

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

11. Applied pow-unpow 1.0

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

13. Applied add-cube-cbrt 1.0

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

14. Applied pow-unpow 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020092 
(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)))