Average Error: 0.5 → 0.4
Time: 21.9s
Precision: 64
Internal Precision: 640
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\begin{array}{l}
\mathbf{if}\;x \le -3.122162421299454 \cdot 10^{-16}:\\
\;\;\;\;\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\right)\\
\mathbf{else}:\\
\;\;\;\;\left(x \cdot \frac{1}{8} + \left(\frac{1}{2} - y\right)\right) \cdot x + \log 2\\
\end{array}\]
Target
| Original | 0.5 |
|---|
| Target | 0.1 |
|---|
| Herbie | 0.4 |
|---|
\[\begin{array}{l}
\mathbf{if}\;x \le 0:\\
\;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\
\mathbf{else}:\\
\;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\
\end{array}\]
Derivation
- Split input into 2 regimes
if x < -3.122162421299454e-16
Initial program 0.1
\[\log \left(1 + e^{x}\right) - x \cdot y\]
- Using strategy
rm Applied add-sqr-sqrt0.2
\[\leadsto \log \color{blue}{\left(\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}\right)} - x \cdot y\]
Applied log-prod0.2
\[\leadsto \color{blue}{\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right)} - x \cdot y\]
Applied associate--l+0.2
\[\leadsto \color{blue}{\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\right)}\]
if -3.122162421299454e-16 < x
Initial program 0.6
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Taylor expanded around 0 0.5
\[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \left(\log 2 + \frac{1}{2} \cdot x\right)\right)} - x \cdot y\]
Applied simplify0.5
\[\leadsto \color{blue}{\left(x \cdot \frac{1}{8} + \left(\frac{1}{2} - y\right)\right) \cdot x + \log 2}\]
- Recombined 2 regimes into one program.
Runtime
herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)'
(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)))
