Average Error: 0.4 → 0.3
Time: 28.0s
Precision: 64
Internal Precision: 896
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\begin{array}{l}
\mathbf{if}\;x \cdot \left((\frac{1}{8} \cdot x + \frac{1}{2})_* - y\right) \le -1.2239097232854343 \cdot 10^{+35}:\\
\;\;\;\;x \cdot \left((\frac{1}{8} \cdot x + \frac{1}{2})_* - y\right)\\
\mathbf{if}\;x \cdot \left((\frac{1}{8} \cdot x + \frac{1}{2})_* - y\right) \le 0.0030382068270067893:\\
\;\;\;\;\frac{{\left(\log_* (1 + e^{x})\right)}^{3} - {\left(y \cdot x\right)}^{3}}{(\left(\log_* (1 + e^{x})\right) \cdot \left(x \cdot y\right) + \left((\left(\log_* (1 + e^{x})\right) \cdot \left(\log_* (1 + e^{x})\right) + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right))_*\right))_*}\\
\mathbf{else}:\\
\;\;\;\;(\left(\sqrt[3]{\log_* (1 + e^{x})} \cdot \sqrt[3]{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt[3]{\log_* (1 + e^{x})}\right) + \left(-y \cdot x\right))_*\\
\end{array}\]
Target
| Original | 0.4 |
|---|
| Target | 0.0 |
|---|
| Herbie | 0.3 |
|---|
\[\begin{array}{l}
\mathbf{if}\;x \le 0:\\
\;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\
\mathbf{else}:\\
\;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\
\end{array}\]
Derivation
- Split input into 3 regimes
if (* x (- (fma 1/8 x 1/2) y)) < -1.2239097232854343e+35
Initial program 1.9
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Applied simplify1.9
\[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]
Taylor expanded around 0 0.8
\[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \left(\log 2 + \frac{1}{2} \cdot x\right)\right)} - y \cdot x\]
Applied simplify0.8
\[\leadsto \color{blue}{(x \cdot \left((x \cdot \frac{1}{8} + \frac{1}{2})_*\right) + \left(\log 2\right))_* - y \cdot x}\]
Taylor expanded around inf 0.8
\[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{2} \cdot x\right) - y \cdot x}\]
Applied simplify0.8
\[\leadsto \color{blue}{x \cdot \left((\frac{1}{8} \cdot x + \frac{1}{2})_* - y\right)}\]
if -1.2239097232854343e+35 < (* x (- (fma 1/8 x 1/2) y)) < 0.0030382068270067893
Initial program 0.0
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Applied simplify0.0
\[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]
- Using strategy
rm Applied flip3--0.0
\[\leadsto \color{blue}{\frac{{\left(\log_* (1 + e^{x})\right)}^{3} - {\left(y \cdot x\right)}^{3}}{\log_* (1 + e^{x}) \cdot \log_* (1 + e^{x}) + \left(\left(y \cdot x\right) \cdot \left(y \cdot x\right) + \log_* (1 + e^{x}) \cdot \left(y \cdot x\right)\right)}}\]
Applied simplify0.1
\[\leadsto \frac{{\left(\log_* (1 + e^{x})\right)}^{3} - {\left(y \cdot x\right)}^{3}}{\color{blue}{(\left(\log_* (1 + e^{x})\right) \cdot \left(x \cdot y\right) + \left((\left(\log_* (1 + e^{x})\right) \cdot \left(\log_* (1 + e^{x})\right) + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right))_*\right))_*}}\]
if 0.0030382068270067893 < (* x (- (fma 1/8 x 1/2) y))
Initial program 0.7
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Applied simplify0.6
\[\leadsto \color{blue}{\log_* (1 + e^{x}) - y \cdot x}\]
- Using strategy
rm Applied add-cube-cbrt0.6
\[\leadsto \color{blue}{\left(\sqrt[3]{\log_* (1 + e^{x})} \cdot \sqrt[3]{\log_* (1 + e^{x})}\right) \cdot \sqrt[3]{\log_* (1 + e^{x})}} - y \cdot x\]
Applied fma-neg0.6
\[\leadsto \color{blue}{(\left(\sqrt[3]{\log_* (1 + e^{x})} \cdot \sqrt[3]{\log_* (1 + e^{x})}\right) \cdot \left(\sqrt[3]{\log_* (1 + e^{x})}\right) + \left(-y \cdot x\right))_*}\]
- Recombined 3 regimes into one program.
Runtime
herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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)))
