Average Error: 0.5 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 896
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \le 3.2703076429549795 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)\\ \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \le 109.39864347721476:\\ \;\;\;\;\frac{{\left(\log \left(1 + e^{x}\right)\right)}^{3} - {\left(x \cdot y\right)}^{3}}{\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right) + \left(y \cdot x\right) \cdot \left(y \cdot x + \log \left(1 + e^{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.1
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)) < 3.2703076429549795e-06 or 109.39864347721476 < (- (log (+ 1 (exp x))) (* x y))

    1. Initial program 1.1

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

    3. Applied add-cube-cbrt1.1

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

    4. Applied log-prod1.1

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

    5. Applied associate--l+1.1

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

    if 3.2703076429549795e-06 < (- (log (+ 1 (exp x))) (* x y)) < 109.39864347721476

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied simplify0.0

      \[\leadsto \frac{{\left(\log \left(1 + e^{x}\right)\right)}^{3} - {\left(x \cdot y\right)}^{3}}{\color{blue}{\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right) + \left(y \cdot x\right) \cdot \left(y \cdot x + \log \left(1 + e^{x}\right)\right)}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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)))