Average Error: 0.4 → 0.4
Time: 5.7s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{{\left(\log \left(e^{x} + 1\right)\right)}^{3}} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{{\left(\log \left(e^{x} + 1\right)\right)}^{3}} - x \cdot y
double f(double x, double y) {
        double r156691 = 1.0;
        double r156692 = x;
        double r156693 = exp(r156692);
        double r156694 = r156691 + r156693;
        double r156695 = log(r156694);
        double r156696 = y;
        double r156697 = r156692 * r156696;
        double r156698 = r156695 - r156697;
        return r156698;
}
↓
double f(double x, double y) {
        double r156699 = x;
        double r156700 = exp(r156699);
        double r156701 = 1.0;
        double r156702 = r156700 + r156701;
        double r156703 = log(r156702);
        double r156704 = 3.0;
        double r156705 = pow(r156703, r156704);
        double r156706 = cbrt(r156705);
        double r156707 = y;
        double r156708 = r156699 * r156707;
        double r156709 = r156706 - r156708;
        return r156709;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  3. Applied add-cbrt-cube0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

    \[\leadsto \sqrt[3]{{\left(\log \left(e^{x} + 1\right)\right)}^{3}} - x \cdot y\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(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)))