Average Error: 0.5 → 1.1
Time: 12.6s
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 r95232 = 1.0;
        double r95233 = x;
        double r95234 = exp(r95233);
        double r95235 = r95232 + r95234;
        double r95236 = log(r95235);
        double r95237 = y;
        double r95238 = r95233 * r95237;
        double r95239 = r95236 - r95238;
        return r95239;
}
↓
double f(double x, double y) {
        double r95240 = 2.0;
        double r95241 = 1.0;
        double r95242 = x;
        double r95243 = exp(r95242);
        double r95244 = r95241 + r95243;
        double r95245 = sqrt(r95244);
        double r95246 = log(r95245);
        double r95247 = r95240 * r95246;
        double r95248 = y;
        double r95249 = r95242 * r95248;
        double r95250 = r95247 - r95249;
        return r95250;
}

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

Derivation

  1. Initial program 0.5

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

  3. Applied add-sqr-sqrt1.4

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

  4. Applied log-prod1.1

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

  5. Final simplification1.1

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

Reproduce

herbie shell --seed 2019298 
(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)))