Average Error: 0.5 → 0.5
Time: 5.0s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}} \cdot \sqrt{\sqrt{1 + e^{x}}}\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}} \cdot \sqrt{\sqrt{1 + e^{x}}}\right)\right) - x \cdot y
double f(double x, double y) {
        double r111428 = 1.0;
        double r111429 = x;
        double r111430 = exp(r111429);
        double r111431 = r111428 + r111430;
        double r111432 = log(r111431);
        double r111433 = y;
        double r111434 = r111429 * r111433;
        double r111435 = r111432 - r111434;
        return r111435;
}
↓
double f(double x, double y) {
        double r111436 = 1.0;
        double r111437 = x;
        double r111438 = exp(r111437);
        double r111439 = r111436 + r111438;
        double r111440 = sqrt(r111439);
        double r111441 = log(r111440);
        double r111442 = sqrt(r111440);
        double r111443 = r111442 * r111442;
        double r111444 = log(r111443);
        double r111445 = r111441 + r111444;
        double r111446 = y;
        double r111447 = r111437 * r111446;
        double r111448 = r111445 - r111447;
        return r111448;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

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. Using strategy rm

  6. Applied add-sqr-sqrt1.0

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

  7. Applied sqrt-prod0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020034 +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)))