Average Error: 0.5 → 0.6

Time: 16.4s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	0.5
Target	0.1
Herbie	0.6

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

Derivation

Initial program 0.5
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Using strategy rm
Applied flip3-+0.6
\[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - x \cdot y\]
Applied log-div0.6
\[\leadsto \color{blue}{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)} - x \cdot y\]
Applied associate--l-0.6
\[\leadsto \color{blue}{\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + x \cdot y\right)}\]
Simplified0.6
\[\leadsto \color{blue}{\log \left(e^{\left(x + x\right) + x} + 1\right)} - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + x \cdot y\right)\]
Final simplification0.6
\[\leadsto \log \left(e^{\left(x + x\right) + x} + 1\right) - \left(x \cdot y + \log \left(\left(e^{x} \cdot e^{x} - e^{x}\right) + 1\right)\right)\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))