Average Error: 5.4 → 0.1
Time: 2.2s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[x \cdot 1 + \left(x \cdot y\right) \cdot y\]
x \cdot \left(1 + y \cdot y\right)
↓
x \cdot 1 + \left(x \cdot y\right) \cdot y
double f(double x, double y) {
        double r455321 = x;
        double r455322 = 1.0;
        double r455323 = y;
        double r455324 = r455323 * r455323;
        double r455325 = r455322 + r455324;
        double r455326 = r455321 * r455325;
        return r455326;
}
↓
double f(double x, double y) {
        double r455327 = x;
        double r455328 = 1.0;
        double r455329 = r455327 * r455328;
        double r455330 = y;
        double r455331 = r455327 * r455330;
        double r455332 = r455331 * r455330;
        double r455333 = r455329 + r455332;
        return r455333;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original5.4
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.4

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

  3. Applied distribute-lft-in5.4

    \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(y \cdot y\right)}\]
  4. Using strategy rm

  5. Applied associate-*r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(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))))