Average Error: 0.5 → 0.6
Time: 5.1s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right) - x \cdot y
double f(double x, double y) {
        double r131963 = 1.0;
        double r131964 = x;
        double r131965 = exp(r131964);
        double r131966 = r131963 + r131965;
        double r131967 = log(r131966);
        double r131968 = y;
        double r131969 = r131964 * r131968;
        double r131970 = r131967 - r131969;
        return r131970;
}
↓
double f(double x, double y) {
        double r131971 = 1.0;
        double r131972 = 3.0;
        double r131973 = pow(r131971, r131972);
        double r131974 = x;
        double r131975 = exp(r131974);
        double r131976 = pow(r131975, r131972);
        double r131977 = r131973 + r131976;
        double r131978 = log(r131977);
        double r131979 = r131971 * r131971;
        double r131980 = r131975 * r131975;
        double r131981 = r131971 * r131975;
        double r131982 = r131980 - r131981;
        double r131983 = r131979 + r131982;
        double r131984 = log(r131983);
        double r131985 = r131978 - r131984;
        double r131986 = y;
        double r131987 = r131974 * r131986;
        double r131988 = r131985 - r131987;
        return r131988;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+0.6

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

  4. Applied log-div0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019354 
(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)))