Average Error: 0.5 → 0.5
Time: 26.7s
Precision: 64
Internal Precision: 896
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \le 4.158861846500903 \cdot 10^{-29}:\\ \;\;\;\;(\left(\sqrt[3]{\log_* (1 + e^{x})} \cdot \sqrt[3]{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt[3]{\log_* (1 + e^{x})}\right) + \left(-y \cdot x\right))_*\\ \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \le 1569156637633.598:\\ \;\;\;\;e^{\log \left(\log_* (1 + e^{x}) - y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\log_* (1 + e^{x})} \cdot \sqrt[3]{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt[3]{\log_* (1 + e^{x})}\right) + \left(-y \cdot x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ 1 (exp x))) (* x y)) < 4.158861846500903e-29 or 1569156637633.598 < (- (log (+ 1 (exp x))) (* x y))

    1. Initial program 1.1

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

    2. Applied simplify1.1

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

    4. Applied add-cube-cbrt1.1

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

    5. Applied fma-neg1.1

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

    if 4.158861846500903e-29 < (- (log (+ 1 (exp x))) (* x y)) < 1569156637633.598

    1. Initial program 0.0

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

    2. Applied simplify0.0

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

    4. Applied add-exp-log0.1

      \[\leadsto \color{blue}{e^{\log \left(\log_* (1 + e^{x}) - y \cdot x\right)}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 26.7s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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)))