Average Error: 5.3 → 0.1
Time: 1.2s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\mathsf{fma}\left(x \cdot y, y, x \cdot 1\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{fma}\left(x \cdot y, y, x \cdot 1\right)
double code(double x, double y) {
	return (x * (1.0 + (y * y)));
}
↓
double code(double x, double y) {
	return fma((x * y), y, (x * 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
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 +-commutative5.3

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

  4. Applied distribute-lft-in5.3

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

  6. Applied associate-*r*0.1

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

  7. Applied fma-def0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020071 +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))))