Average Error: 0.5 → 0.5
Time: 16.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(e^{x} + 1\right) - y \cdot x\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(e^{x} + 1\right) - y \cdot x
double f(double x, double y) {
        double r7162283 = 1.0;
        double r7162284 = x;
        double r7162285 = exp(r7162284);
        double r7162286 = r7162283 + r7162285;
        double r7162287 = log(r7162286);
        double r7162288 = y;
        double r7162289 = r7162284 * r7162288;
        double r7162290 = r7162287 - r7162289;
        return r7162290;
}
↓
double f(double x, double y) {
        double r7162291 = x;
        double r7162292 = exp(r7162291);
        double r7162293 = 1.0;
        double r7162294 = r7162292 + r7162293;
        double r7162295 = log(r7162294);
        double r7162296 = y;
        double r7162297 = r7162296 * r7162291;
        double r7162298 = r7162295 - r7162297;
        return r7162298;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around inf 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019168 
(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)))