Average Error: 5.3 → 0.1
Time: 14.0s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[x \cdot 1 + y \cdot \left(x \cdot y\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
x \cdot 1 + y \cdot \left(x \cdot y\right)
double f(double x, double y) {
        double r624614 = x;
        double r624615 = 1.0;
        double r624616 = y;
        double r624617 = r624616 * r624616;
        double r624618 = r624615 + r624617;
        double r624619 = r624614 * r624618;
        return r624619;
}
↓
double f(double x, double y) {
        double r624620 = x;
        double r624621 = 1.0;
        double r624622 = r624620 * r624621;
        double r624623 = y;
        double r624624 = r624620 * r624623;
        double r624625 = r624623 * r624624;
        double r624626 = r624622 + r624625;
        return r624626;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original5.3
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.3

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

  3. Applied distribute-lft-in5.3

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

  4. Simplified5.3

    \[\leadsto x \cdot 1 + \color{blue}{{y}^{2} \cdot x}\]
  5. Using strategy rm

  6. Applied unpow25.3

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

  7. Applied associate-*l*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 +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))))