Average Error: 0.6 → 0.6
Time: 10.8s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}}\right)\right)\right) - y \cdot x\]
\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) - y \cdot x
double f(double x, double y) {
        double r3178463 = 1.0;
        double r3178464 = x;
        double r3178465 = exp(r3178464);
        double r3178466 = r3178463 + r3178465;
        double r3178467 = log(r3178466);
        double r3178468 = y;
        double r3178469 = r3178464 * r3178468;
        double r3178470 = r3178467 - r3178469;
        return r3178470;
}
↓
double f(double x, double y) {
        double r3178471 = 1.0;
        double r3178472 = x;
        double r3178473 = exp(r3178472);
        double r3178474 = r3178471 + r3178473;
        double r3178475 = sqrt(r3178474);
        double r3178476 = log(r3178475);
        double r3178477 = sqrt(r3178475);
        double r3178478 = log(r3178477);
        double r3178479 = r3178478 + r3178478;
        double r3178480 = r3178476 + r3178479;
        double r3178481 = y;
        double r3178482 = r3178481 * r3178472;
        double r3178483 = r3178480 - r3178482;
        return r3178483;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

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

  3. Applied add-sqr-sqrt1.4

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

  4. Applied log-prod1.1

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

  6. Applied add-sqr-sqrt1.1

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

  7. Applied sqrt-prod0.6

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

  8. Applied log-prod0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019156 
(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)))