Average Error: 0.5 → 0.5

Time: 16.7s

Precision: 64

Internal Precision: 832

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

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In
Out

Enter valid numbers for all inputs

Target

Original	0.5
Target	0.1
Herbie	0.5

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

Derivation

Initial program 0.5
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Taylor expanded around inf 0.5
\[\leadsto \log \color{blue}{\left(e^{x} + 1\right)} - x \cdot y\]
Using strategy rm
Applied flip3-+0.5
\[\leadsto \log \color{blue}{\left(\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}\right)} - x \cdot y\]
Applied log-div0.5
\[\leadsto \color{blue}{\left(\log \left({\left(e^{x}\right)}^{3} + {1}^{3}\right) - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right)} - x \cdot y\]
Simplified0.5
\[\leadsto \left(\color{blue}{\log \left(1 + {\left(e^{x}\right)}^{3}\right)} - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right) - x \cdot y\]
Taylor expanded around -inf 0.5
\[\leadsto \left(\color{blue}{\log \left(e^{3 \cdot x} + 1\right)} - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right) - x \cdot y\]
Final simplification0.5
\[\leadsto \left(\log \left(e^{3 \cdot x} + 1\right) - \log \left(e^{x} \cdot e^{x} + \left(1 - e^{x}\right)\right)\right) - y \cdot x\]

Runtime

herbie shell --seed 2018230 
(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)))