Average Error: 0.5 → 1.0
Time: 10.7s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[2 \cdot \log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
2 \cdot \log \left(\sqrt{1 + e^{x}}\right) - x \cdot y
double f(double x, double y) {
        double r139302 = 1.0;
        double r139303 = x;
        double r139304 = exp(r139303);
        double r139305 = r139302 + r139304;
        double r139306 = log(r139305);
        double r139307 = y;
        double r139308 = r139303 * r139307;
        double r139309 = r139306 - r139308;
        return r139309;
}
↓
double f(double x, double y) {
        double r139310 = 2.0;
        double r139311 = 1.0;
        double r139312 = x;
        double r139313 = exp(r139312);
        double r139314 = r139311 + r139313;
        double r139315 = sqrt(r139314);
        double r139316 = log(r139315);
        double r139317 = r139310 * r139316;
        double r139318 = y;
        double r139319 = r139312 * r139318;
        double r139320 = r139317 - r139319;
        return r139320;
}

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. 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

    \[\leadsto 2 \cdot \log \left(\sqrt{1 + e^{x}}\right) - x \cdot y\]

Reproduce

herbie shell --seed 2019297 
(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)))