Average Error: 0.5 → 1.0
Time: 14.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + e^{\log \left(\log \left(\sqrt{1 + e^{x}}\right)\right)}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{1 + e^{x}}\right) + e^{\log \left(\log \left(\sqrt{1 + e^{x}}\right)\right)}\right) - x \cdot y
double f(double x, double y) {
        double r166335 = 1.0;
        double r166336 = x;
        double r166337 = exp(r166336);
        double r166338 = r166335 + r166337;
        double r166339 = log(r166338);
        double r166340 = y;
        double r166341 = r166336 * r166340;
        double r166342 = r166339 - r166341;
        return r166342;
}
↓
double f(double x, double y) {
        double r166343 = 1.0;
        double r166344 = x;
        double r166345 = exp(r166344);
        double r166346 = r166343 + r166345;
        double r166347 = sqrt(r166346);
        double r166348 = log(r166347);
        double r166349 = log(r166348);
        double r166350 = exp(r166349);
        double r166351 = r166348 + r166350;
        double r166352 = y;
        double r166353 = r166344 * r166352;
        double r166354 = r166351 - r166353;
        return r166354;
}

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. Simplified0.5

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

  4. Applied add-sqr-sqrt1.4

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

  5. Applied log-prod1.0

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

  7. Applied add-exp-log1.0

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

  8. Final simplification1.0

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))