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Target

Derivation

1. Split input into 2 regimes

2. if (- (log (+ 1 (exp x))) (* x y)) < 0.49429716277312463 or 31.001698259014866 < (- (log (+ 1 (exp x))) (* x y))

   1. Initial program 0.9

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

   3. Applied add-cube-cbrt0.9

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

   4. Applied log-prod0.9

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

   if 0.49429716277312463 < (- (log (+ 1 (exp x))) (* x y)) < 31.001698259014866

   1. Initial program 0.0

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

   3. Applied add-log-exp0.0

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

   4. Applied diff-log0.0

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

4. Applied simplify0.4

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - y \cdot x \le 0.49429716277312463 \lor \neg \left(\log \left(1 + e^{x}\right) - y \cdot x \le 31.001698259014866\right):\\ \;\;\;\;\left(\log \left(\sqrt[3]{1 + e^{x}}\right) + \log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right)\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + e^{x}}{e^{y \cdot x}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018207 
(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)))