Average Error: 5.3 → 5.3
Time: 12.2s
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 r297909 = x;
        double r297910 = 1.0;
        double r297911 = y;
        double r297912 = r297911 * r297911;
        double r297913 = r297910 + r297912;
        double r297914 = r297909 * r297913;
        return r297914;
}
↓
double f(double x, double y) {
        double r297915 = x;
        double r297916 = 1.0;
        double r297917 = y;
        double r297918 = r297917 * r297917;
        double r297919 = r297916 + r297918;
        double r297920 = sqrt(r297919);
        double r297921 = r297915 * r297920;
        double r297922 = r297921 * r297920;
        return r297922;
}

Error

Bits error versus x

Bits error versus y

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Results

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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 2019199 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

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

  (* x (+ 1.0 (* y y))))