Average Error: 5.6 → 5.6
Time: 2.9s
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 x * (1.0 + (y * y));
}
↓
double code(double x, double y) {
	return sqrt(1.0 + (y * y)) * (x * sqrt(1.0 + (y * y)));
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.6
Target0.1
Herbie5.6
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.6

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

  3. Applied add-sqr-sqrt_binary64_130585.6

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

  4. Applied associate-*r*_binary64_129765.6

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

  5. Final simplification5.6

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

Reproduce

herbie shell --seed 2020295 
(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))))