Average Error: 5.3 → 5.3
Time: 3.1s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\left(x \cdot \sqrt{1 + y \cdot y}\right) \cdot \sqrt{1 + y \cdot y}\]
x \cdot \left(1 + y \cdot y\right)
↓
\left(x \cdot \sqrt{1 + y \cdot y}\right) \cdot \sqrt{1 + y \cdot y}
double f(double x, double y) {
        double r550617 = x;
        double r550618 = 1.0;
        double r550619 = y;
        double r550620 = r550619 * r550619;
        double r550621 = r550618 + r550620;
        double r550622 = r550617 * r550621;
        return r550622;
}
↓
double f(double x, double y) {
        double r550623 = x;
        double r550624 = 1.0;
        double r550625 = y;
        double r550626 = r550625 * r550625;
        double r550627 = r550624 + r550626;
        double r550628 = sqrt(r550627);
        double r550629 = r550623 * r550628;
        double r550630 = r550629 * r550628;
        return r550630;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.3
Target0.1
Herbie5.3
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.3

    \[x \cdot \left(1 + y \cdot y\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt5.3

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

  4. Applied associate-*r*5.3

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

  5. Final simplification5.3

    \[\leadsto \left(x \cdot \sqrt{1 + y \cdot y}\right) \cdot \sqrt{1 + y \cdot y}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

  (* x (+ 1 (* y y))))