Average Error: 5.4 → 5.5
Time: 2.2s
Precision: binary64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\sqrt{1 + y \cdot y} \cdot \left(x \cdot \sqrt{1 + y \cdot y}\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
\sqrt{1 + y \cdot y} \cdot \left(x \cdot \sqrt{1 + y \cdot y}\right)
(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (* (sqrt (+ 1.0 (* y y))) (* x (sqrt (+ 1.0 (* y 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) sqrt(((double) (1.0 + ((double) (y * y)))))) * ((double) (x * ((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

Original5.4
Target0.1
Herbie5.5
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.4

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

  3. Applied add-sqr-sqrt5.5

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

  4. Applied associate-*r*5.5

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

  5. Final simplification5.5

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

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

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