Average Error: 5.3 → 0.1
Time: 13.7s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[1 \cdot x + y \cdot \left(x \cdot y\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
1 \cdot x + y \cdot \left(x \cdot y\right)
double f(double x, double y) {
        double r639054 = x;
        double r639055 = 1.0;
        double r639056 = y;
        double r639057 = r639056 * r639056;
        double r639058 = r639055 + r639057;
        double r639059 = r639054 * r639058;
        return r639059;
}
↓
double f(double x, double y) {
        double r639060 = 1.0;
        double r639061 = x;
        double r639062 = r639060 * r639061;
        double r639063 = y;
        double r639064 = r639061 * r639063;
        double r639065 = r639063 * r639064;
        double r639066 = r639062 + r639065;
        return r639066;
}

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

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

  5. Simplified5.3

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

  7. Applied unpow25.3

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

  8. Applied associate-*l*0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 
(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))))