Average Error: 0.5 → 0.6

Time: 16.9s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

Target

Original	0.5
Target	0.0
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 add-cbrt-cube0.6
\[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)\right) \cdot \log \left(1 + e^{x}\right)}} - x \cdot y\]
Final simplification0.6
\[\leadsto \sqrt[3]{\log \left(1 + e^{x}\right) \cdot \left(\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)\right)} - y \cdot x\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(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)))