Average Error: 5.3 → 0.1
Time: 12.2s
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 r1081547 = x;
        double r1081548 = 1.0;
        double r1081549 = y;
        double r1081550 = r1081549 * r1081549;
        double r1081551 = r1081548 + r1081550;
        double r1081552 = r1081547 * r1081551;
        return r1081552;
}
↓
double f(double x, double y) {
        double r1081553 = 1.0;
        double r1081554 = x;
        double r1081555 = r1081553 * r1081554;
        double r1081556 = y;
        double r1081557 = r1081554 * r1081556;
        double r1081558 = r1081557 * r1081556;
        double r1081559 = r1081555 + r1081558;
        return r1081559;
}

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. 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 2020047 
(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))))