Average Error: 0.5 → 1.0
Time: 18.2s
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 r6241543 = 1.0;
        double r6241544 = x;
        double r6241545 = exp(r6241544);
        double r6241546 = r6241543 + r6241545;
        double r6241547 = log(r6241546);
        double r6241548 = y;
        double r6241549 = r6241544 * r6241548;
        double r6241550 = r6241547 - r6241549;
        return r6241550;
}
↓
double f(double x, double y) {
        double r6241551 = 1.0;
        double r6241552 = x;
        double r6241553 = exp(r6241552);
        double r6241554 = r6241551 + r6241553;
        double r6241555 = sqrt(r6241554);
        double r6241556 = log(r6241555);
        double r6241557 = y;
        double r6241558 = r6241557 * r6241552;
        double r6241559 = r6241556 - r6241558;
        double r6241560 = r6241556 + r6241559;
        return r6241560;
}

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
Target0.0
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 2019163 +o rules:numerics
(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)))