Average Error: 5.3 → 0.1
Time: 14.3s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[x \cdot 1 + \left(x \cdot y\right) \cdot y\]
x \cdot \left(1 + y \cdot y\right)
↓
x \cdot 1 + \left(x \cdot y\right) \cdot y
double f(double x, double y) {
        double r564959 = x;
        double r564960 = 1.0;
        double r564961 = y;
        double r564962 = r564961 * r564961;
        double r564963 = r564960 + r564962;
        double r564964 = r564959 * r564963;
        return r564964;
}
↓
double f(double x, double y) {
        double r564965 = x;
        double r564966 = 1.0;
        double r564967 = r564965 * r564966;
        double r564968 = y;
        double r564969 = r564965 * r564968;
        double r564970 = r564969 * r564968;
        double r564971 = r564967 + r564970;
        return r564971;
}

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

    \[\leadsto x \cdot 1 + \color{blue}{\left(y \cdot y\right) \cdot x}\]
  5. Using strategy rm

  6. Applied pow15.3

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

  7. Applied pow15.3

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

  8. Applied pow15.3

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

  9. Applied pow-prod-down5.3

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

  10. Applied pow-prod-down5.3

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

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))))