Average Error: 5.2 → 5.2
Time: 3.2s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\left(x \cdot \sqrt{1 + y \cdot y}\right) \cdot \sqrt{1 + y \cdot y}\]
x \cdot \left(1 + y \cdot y\right)
↓
\left(x \cdot \sqrt{1 + y \cdot y}\right) \cdot \sqrt{1 + y \cdot y}
double code(double x, double y) {
	return ((double) (x * ((double) (1.0 + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (x * ((double) sqrt(((double) (1.0 + ((double) (y * y)))))))) * ((double) sqrt(((double) (1.0 + ((double) (y * y))))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.2
Target0.1
Herbie5.2
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.2

    \[x \cdot \left(1 + y \cdot y\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt5.2

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

  4. Applied associate-*r*5.2

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

  5. Final simplification5.2

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

Reproduce

herbie shell --seed 2020113 +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))))