Average Error: 4.9 → 4.9
Time: 8.9s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\left(1 + y \cdot y\right) \cdot x\]
x \cdot \left(1 + y \cdot y\right)
↓
\left(1 + y \cdot y\right) \cdot x
double f(double x, double y) {
        double r317153 = x;
        double r317154 = 1.0;
        double r317155 = y;
        double r317156 = r317155 * r317155;
        double r317157 = r317154 + r317156;
        double r317158 = r317153 * r317157;
        return r317158;
}
↓
double f(double x, double y) {
        double r317159 = 1.0;
        double r317160 = y;
        double r317161 = r317160 * r317160;
        double r317162 = r317159 + r317161;
        double r317163 = x;
        double r317164 = r317162 * r317163;
        return r317164;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original4.9
Target0.1
Herbie4.9
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 4.9

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

  2. Simplified4.9

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

  3. Final simplification4.9

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

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

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