Average Error: 5.3 → 5.3
Time: 1.4s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[x \cdot \mathsf{fma}\left(y, y, 1\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
x \cdot \mathsf{fma}\left(y, y, 1\right)
double code(double x, double y) {
	return (x * (1.0 + (y * y)));
}
↓
double code(double x, double y) {
	return (x * fma(y, y, 1.0));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around inf 5.3

    \[\leadsto \color{blue}{x \cdot {y}^{2} + 1 \cdot x}\]

  3. Simplified5.3

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

  4. Final simplification5.3

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

Reproduce

herbie shell --seed 2020092 +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))))