Average Error: 0.5 → 0.5
Time: 13.4s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{2} \cdot e^{x}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{2} \cdot e^{x}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y
double f(double x, double y) {
        double r108992 = 1.0;
        double r108993 = x;
        double r108994 = exp(r108993);
        double r108995 = r108992 + r108994;
        double r108996 = log(r108995);
        double r108997 = y;
        double r108998 = r108993 * r108997;
        double r108999 = r108996 - r108998;
        return r108999;
}
↓
double f(double x, double y) {
        double r109000 = 1.0;
        double r109001 = 3.0;
        double r109002 = pow(r109000, r109001);
        double r109003 = x;
        double r109004 = exp(r109003);
        double r109005 = 2.0;
        double r109006 = pow(r109004, r109005);
        double r109007 = r109006 * r109004;
        double r109008 = r109002 + r109007;
        double r109009 = r109004 - r109000;
        double r109010 = r109004 * r109009;
        double r109011 = r109000 * r109000;
        double r109012 = r109010 + r109011;
        double r109013 = r109008 / r109012;
        double r109014 = log(r109013);
        double r109015 = y;
        double r109016 = r109003 * r109015;
        double r109017 = r109014 - r109016;
        return r109017;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+0.5

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

  4. Simplified0.5

    \[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{\color{blue}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}}\right) - x \cdot y\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.6

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

  7. Applied unpow-prod-down0.6

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

  8. Simplified0.5

    \[\leadsto \log \left(\frac{{1}^{3} + \color{blue}{{\left(e^{x}\right)}^{2}} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{3}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y\]

  9. Simplified0.5

    \[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{2} \cdot \color{blue}{e^{x}}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y\]

  10. Final simplification0.5

    \[\leadsto \log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{2} \cdot e^{x}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y\]

Reproduce

herbie shell --seed 2019326 
(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)))