Average Error: 0.5 → 1.0
Time: 31.1s
Precision: 64
Internal Precision: 128
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[(\left(\sqrt{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt{\log_* (1 + e^{x})}\right) + \left(x \cdot \left(-y\right)\right))_*\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.1
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied fma-neg1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2019091 +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)))