Average Error: 0.4 → 0.4
Time: 12.1s
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 r117479 = 1.0;
        double r117480 = x;
        double r117481 = exp(r117480);
        double r117482 = r117479 + r117481;
        double r117483 = log(r117482);
        double r117484 = y;
        double r117485 = r117480 * r117484;
        double r117486 = r117483 - r117485;
        return r117486;
}
↓
double f(double x, double y) {
        double r117487 = 1.0;
        double r117488 = x;
        double r117489 = exp(r117488);
        double r117490 = r117487 + r117489;
        double r117491 = log(r117490);
        double r117492 = sqrt(r117491);
        double r117493 = sqrt(r117490);
        double r117494 = log(r117493);
        double r117495 = r117494 + r117494;
        double r117496 = sqrt(r117495);
        double r117497 = r117492 * r117496;
        double r117498 = y;
        double r117499 = r117488 * r117498;
        double r117500 = r117497 - r117499;
        return r117500;
}

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 
(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)))