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Target

Derivation

1. Initial program 0.5

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

2. Initial simplification0.5

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

4. Applied flip3-+0.6

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

5. Applied log-div0.6

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

6. Applied associate--l-0.6

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

7. Simplified0.6

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

8. Final simplification0.6

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

Runtime

Time bar (total: 17.3s)Debug logProfile

herbie shell --seed 2018274 
(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)))