Average Error: 5.3 → 0.1
Time: 9.7s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[1 \cdot x + \left(x \cdot y\right) \cdot y\]
x \cdot \left(1 + y \cdot y\right)
↓
1 \cdot x + \left(x \cdot y\right) \cdot y
double f(double x, double y) {
        double r390014 = x;
        double r390015 = 1.0;
        double r390016 = y;
        double r390017 = r390016 * r390016;
        double r390018 = r390015 + r390017;
        double r390019 = r390014 * r390018;
        return r390019;
}
↓
double f(double x, double y) {
        double r390020 = 1.0;
        double r390021 = x;
        double r390022 = r390020 * r390021;
        double r390023 = y;
        double r390024 = r390021 * r390023;
        double r390025 = r390024 * r390023;
        double r390026 = r390022 + r390025;
        return r390026;
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

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. Using strategy rm

  6. Applied associate-*r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(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))))