Average Error: 0.4 → 0.4
Time: 11.7s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt{\log \left(1 + e^{x}\right)} \cdot \sqrt{\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt{\log \left(1 + e^{x}\right)} \cdot \sqrt{\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)} - x \cdot y
double f(double x, double y) {
        double r99392 = 1.0;
        double r99393 = x;
        double r99394 = exp(r99393);
        double r99395 = r99392 + r99394;
        double r99396 = log(r99395);
        double r99397 = y;
        double r99398 = r99393 * r99397;
        double r99399 = r99396 - r99398;
        return r99399;
}
↓
double f(double x, double y) {
        double r99400 = 1.0;
        double r99401 = x;
        double r99402 = exp(r99401);
        double r99403 = r99400 + r99402;
        double r99404 = log(r99403);
        double r99405 = sqrt(r99404);
        double r99406 = sqrt(r99403);
        double r99407 = log(r99406);
        double r99408 = r99407 + r99407;
        double r99409 = sqrt(r99408);
        double r99410 = r99405 * r99409;
        double r99411 = y;
        double r99412 = r99401 * r99411;
        double r99413 = r99410 - r99412;
        return r99413;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  3. Applied add-sqr-sqrt0.9

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

  5. Applied add-sqr-sqrt0.9

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

  6. Applied log-prod0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))