Average Error: 0.5 → 0.5
Time: 26.5s
Precision: 64
Internal Precision: 576
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}}\right)\right)\right) - x \cdot y\]

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. Initial program 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt1.3

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

  4. Applied log-prod1.0

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

  6. Applied add-sqr-sqrt1.0

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

  7. Applied sqrt-prod0.5

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

  8. Applied log-prod0.5

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

Runtime

Time bar (total: 26.5s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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)))