Average Error: 0.5 → 1.1
Time: 13.0s
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 r133958 = 1.0;
        double r133959 = x;
        double r133960 = exp(r133959);
        double r133961 = r133958 + r133960;
        double r133962 = log(r133961);
        double r133963 = y;
        double r133964 = r133959 * r133963;
        double r133965 = r133962 - r133964;
        return r133965;
}
↓
double f(double x, double y) {
        double r133966 = 2.0;
        double r133967 = 1.0;
        double r133968 = x;
        double r133969 = exp(r133968);
        double r133970 = r133967 + r133969;
        double r133971 = sqrt(r133970);
        double r133972 = log(r133971);
        double r133973 = r133966 * r133972;
        double r133974 = y;
        double r133975 = r133968 * r133974;
        double r133976 = r133973 - r133975;
        return r133976;
}

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

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. Applied associate--l+1.1

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

  6. Final simplification1.1

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

Reproduce

herbie shell --seed 2019294 
(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)))