Average Error: 0.5 → 0.5
Time: 20.1s
Precision: 64
Internal precision: 128
\[\log \left(1 + e^{x}\right) - x \cdot y\]
⬇
\[\left(\log \left({\left(e^{x}\right)}^3 + 1\right) - \log \left({\left(e^{x}\right)}^2 - \left(e^{x} - 1\right)\right)\right) - x \cdot y\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Comparison0.1
Herbie0.5
\[ \begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+ 0.5

    \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{{1}^2 + \left({\left(e^{x}\right)}^2 - 1 \cdot e^{x}\right)}\right)} - x \cdot y\]

  4. Applied log-div 0.5

    \[\leadsto \color{blue}{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left({1}^2 + \left({\left(e^{x}\right)}^2 - 1 \cdot e^{x}\right)\right)\right)} - x \cdot y\]

  5. Applied simplify 0.5

    \[\leadsto \left(\color{blue}{\log \left({\left(e^{x}\right)}^3 + 1\right)} - \log \left({1}^2 + \left({\left(e^{x}\right)}^2 - 1 \cdot e^{x}\right)\right)\right) - x \cdot y\]

  6. Applied simplify 0.5

    \[\leadsto \left(\log \left({\left(e^{x}\right)}^3 + 1\right) - \color{blue}{\log \left({\left(e^{x}\right)}^2 - \left(e^{x} - 1\right)\right)}\right) - x \cdot y\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 20.1s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2360045958 3893460442 2224999912 1019979277 4208983693 2410681664)'
(FPCore (x y)
  :name "Logistic regression 2"

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

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