Average Error: 0.5 → 1.0
Time: 4.3s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y
double f(double x, double y) {
        double r157546 = 1.0;
        double r157547 = x;
        double r157548 = exp(r157547);
        double r157549 = r157546 + r157548;
        double r157550 = log(r157549);
        double r157551 = y;
        double r157552 = r157547 * r157551;
        double r157553 = r157550 - r157552;
        return r157553;
}
↓
double f(double x, double y) {
        double r157554 = 1.0;
        double r157555 = x;
        double r157556 = exp(r157555);
        double r157557 = r157554 + r157556;
        double r157558 = sqrt(r157557);
        double r157559 = log(r157558);
        double r157560 = r157559 + r157559;
        double r157561 = y;
        double r157562 = r157555 * r157561;
        double r157563 = r157560 - r157562;
        return r157563;
}

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
Target0.1
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  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. Final simplification1.0

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

Reproduce

herbie shell --seed 2020020 
(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)))