Average Error: 0.5 → 0.5
Time: 25.4s
Precision: 64
Internal Precision: 640
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(1 + {\left(e^{x}\right)}^{3}\right) - \log \left(\left(1 - e^{x}\right) + e^{x + x}\right)\right) - x \cdot y\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.1
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

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

  4. Applied log-div0.5

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

  5. Applied simplify0.5

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

  6. Applied simplify0.5

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

  7. Applied simplify0.5

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

Runtime

Time bar (total: 25.4s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))