Average Error: 5.3 → 5.3
Time: 7.7s
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 r463200 = x;
        double r463201 = 1.0;
        double r463202 = y;
        double r463203 = r463202 * r463202;
        double r463204 = r463201 + r463203;
        double r463205 = r463200 * r463204;
        return r463205;
}
↓
double f(double x, double y) {
        double r463206 = 1.0;
        double r463207 = y;
        double r463208 = r463207 * r463207;
        double r463209 = r463206 + r463208;
        double r463210 = x;
        double r463211 = r463209 * r463210;
        return r463211;
}

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. Simplified5.3

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

  3. Final simplification5.3

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

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

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