Average Error: 0.5 → 0.6
Time: 58.0s
Precision: 64
Internal Precision: 576
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \left(\left(\log \left({\left(e^{x} \cdot e^{x}\right)}^{3} - {\left(e^{x} - 1\right)}^{3}\right) - \log \left(\left(\left(e^{x} - 1\right) + e^{x + x}\right) \cdot e^{x + x} + \left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right)\right) + y \cdot x\right)\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original0.5
Target0.0
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

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 associate--l-0.5

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

  6. Applied simplify0.5

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

  8. Applied flip3--0.6

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

  9. Applied log-div0.6

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

  10. Applied simplify0.6

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

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed 2018208 
(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)))