Average Error: 0.5 → 1.0
Time: 22.5s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - y \cdot x\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{1 + e^{x}}\right) - y \cdot x\right)
double f(double x, double y) {
        double r28651279 = 1.0;
        double r28651280 = x;
        double r28651281 = exp(r28651280);
        double r28651282 = r28651279 + r28651281;
        double r28651283 = log(r28651282);
        double r28651284 = y;
        double r28651285 = r28651280 * r28651284;
        double r28651286 = r28651283 - r28651285;
        return r28651286;
}
↓
double f(double x, double y) {
        double r28651287 = 1.0;
        double r28651288 = x;
        double r28651289 = exp(r28651288);
        double r28651290 = r28651287 + r28651289;
        double r28651291 = sqrt(r28651290);
        double r28651292 = log(r28651291);
        double r28651293 = y;
        double r28651294 = r28651293 * r28651288;
        double r28651295 = r28651292 - r28651294;
        double r28651296 = r28651292 + r28651295;
        return r28651296;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.1
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

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. Applied associate--l+1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2019125 
(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)))