Average Error: 0.5 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 640
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\frac{1 + {\left(e^{x}\right)}^{3}}{e^{x + x} + \left(1 - e^{x}\right)}\right) - x \cdot y\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.0
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

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

  4. Applied simplify0.5

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

  5. Applied simplify0.5

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

Runtime

Time bar (total: 1.0m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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)))