Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.1 → 5.1

Time: 13.0s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(y, y, 1\right) \cdot x\]

C

double f(double x, double y) {
        double r22067415 = x;
        double r22067416 = 1.0;
        double r22067417 = y;
        double r22067418 = r22067417 * r22067417;
        double r22067419 = r22067416 + r22067418;
        double r22067420 = r22067415 * r22067419;
        return r22067420;
}

↓

double f(double x, double y) {
        double r22067421 = y;
        double r22067422 = 1.0;
        double r22067423 = fma(r22067421, r22067421, r22067422);
        double r22067424 = x;
        double r22067425 = r22067423 * r22067424;
        return r22067425;
}

Error

Bits error versus x

Bits error versus y

Target

Original	5.1
Target	0.1
Herbie	5.1

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

Derivation

1. Initial program 5.1

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

2. Simplified 5.1

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

3. Final simplification 5.1

   \[\leadsto \mathsf{fma}\left(y, y, 1\right) \cdot x\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

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

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