Average Error: 0.6 → 0.6
Time: 4.9s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{e^{x} \cdot \left(e^{x} - 1\right) + 1 \cdot 1}\right) - x \cdot y
double f(double x, double y) {
        double r217424 = 1.0;
        double r217425 = x;
        double r217426 = exp(r217425);
        double r217427 = r217424 + r217426;
        double r217428 = log(r217427);
        double r217429 = y;
        double r217430 = r217425 * r217429;
        double r217431 = r217428 - r217430;
        return r217431;
}
↓
double f(double x, double y) {
        double r217432 = 1.0;
        double r217433 = 3.0;
        double r217434 = pow(r217432, r217433);
        double r217435 = x;
        double r217436 = exp(r217435);
        double r217437 = pow(r217436, r217433);
        double r217438 = r217434 + r217437;
        double r217439 = r217436 - r217432;
        double r217440 = r217436 * r217439;
        double r217441 = r217432 * r217432;
        double r217442 = r217440 + r217441;
        double r217443 = r217438 / r217442;
        double r217444 = log(r217443);
        double r217445 = y;
        double r217446 = r217435 * r217445;
        double r217447 = r217444 - r217446;
        return r217447;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.1
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 0.6

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  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. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 
(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)))