Average Error: 0.5 → 0.4
Time: 24.8s
Precision: 64
Internal Precision: 576
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\frac{(y \cdot x + \left(\log_* (1 + e^{x})\right))_*}{\frac{\log_* (1 + e^{x}) + y \cdot x}{\log_* (1 + e^{x}) - y \cdot x}}\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Initial simplification0.4

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

  4. Applied flip--9.4

    \[\leadsto \color{blue}{\frac{\log_* (1 + e^{x}) \cdot \log_* (1 + e^{x}) - \left(y \cdot x\right) \cdot \left(y \cdot x\right)}{\log_* (1 + e^{x}) + y \cdot x}}\]
  5. Using strategy rm

  6. Applied difference-of-squares9.3

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

  7. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{\log_* (1 + e^{x}) + y \cdot x}{\frac{\log_* (1 + e^{x}) + y \cdot x}{\log_* (1 + e^{x}) - y \cdot x}}}\]

  8. Simplified0.4

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

  9. Final simplification0.4

    \[\leadsto \frac{(y \cdot x + \left(\log_* (1 + e^{x})\right))_*}{\frac{\log_* (1 + e^{x}) + y \cdot x}{\log_* (1 + e^{x}) - y \cdot x}}\]

Runtime

Time bar (total: 24.8s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(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)))