Average Error: 0.4 → 0.9
Time: 21.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt{\log \left(\sqrt{1 + e^{x}}\right)} \cdot \sqrt{\log \left(\sqrt{1 + e^{x}}\right)} + \left(\log \left(\sqrt{1 + e^{x}}\right) - y \cdot x\right)\]
double f(double x, double y) {
        double r17750443 = 1.0;
        double r17750444 = x;
        double r17750445 = exp(r17750444);
        double r17750446 = r17750443 + r17750445;
        double r17750447 = log(r17750446);
        double r17750448 = y;
        double r17750449 = r17750444 * r17750448;
        double r17750450 = r17750447 - r17750449;
        return r17750450;
}
↓
double f(double x, double y) {
        double r17750451 = 1.0;
        double r17750452 = x;
        double r17750453 = exp(r17750452);
        double r17750454 = r17750451 + r17750453;
        double r17750455 = sqrt(r17750454);
        double r17750456 = log(r17750455);
        double r17750457 = sqrt(r17750456);
        double r17750458 = r17750457 * r17750457;
        double r17750459 = y;
        double r17750460 = r17750459 * r17750452;
        double r17750461 = r17750456 - r17750460;
        double r17750462 = r17750458 + r17750461;
        return r17750462;
}


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

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.1
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  7. Applied add-sqr-sqrt0.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2019102 
(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)))