Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -3052060746915.3

   1. Initial program 0

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

   3. Applied add-sqr-sqrt0

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

   4. Applied log-prod0

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

   if -3052060746915.3 < x

   1. Initial program 0.6

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

   2. Taylor expanded around 0 1.2

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

   3. Simplified1.2

      \[\leadsto \log \color{blue}{\left((\left((x \cdot \frac{1}{2} + 1)_*\right) \cdot x + 2)_*\right)} - x \cdot y\]
3. Recombined 2 regimes into one program.

4. Final simplification0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3052060746915.3:\\ \;\;\;\;\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left((\left((x \cdot \frac{1}{2} + 1)_*\right) \cdot x + 2)_*\right) - y \cdot x\\ \end{array}\]

Runtime

Time bar (total: 17.9s)Debug logProfile

herbie shell --seed 2018346 +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)))