Average Error: 0.5 → 0.5

Time: 45.3s

Precision: 64

Internal Precision: 576

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

↓

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

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
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Initial simplification0.5
\[\leadsto \log \left(1 + e^{x}\right) - y \cdot x\]
Using strategy rm
Applied flip--9.4
\[\leadsto \color{blue}{\frac{\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right) - \left(y \cdot x\right) \cdot \left(y \cdot x\right)}{\log \left(1 + e^{x}\right) + y \cdot x}}\]
Using strategy rm
Applied difference-of-squares9.4
\[\leadsto \frac{\color{blue}{\left(\log \left(1 + e^{x}\right) + y \cdot x\right) \cdot \left(\log \left(1 + e^{x}\right) - y \cdot x\right)}}{\log \left(1 + e^{x}\right) + y \cdot x}\]
Applied associate-/l*0.5
\[\leadsto \color{blue}{\frac{\log \left(1 + e^{x}\right) + y \cdot x}{\frac{\log \left(1 + e^{x}\right) + y \cdot x}{\log \left(1 + e^{x}\right) - y \cdot x}}}\]
Final simplification0.5
\[\leadsto \frac{y \cdot x + \log \left(1 + e^{x}\right)}{\frac{y \cdot x + \log \left(1 + e^{x}\right)}{\log \left(1 + e^{x}\right) - y \cdot x}}\]

Runtime

herbie shell --seed 2018227 
(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)))