Average Error: 5.3 → 0.1
Time: 5.0s
Precision: binary64
\[x \cdot \left(1 + y \cdot y\right) \]
↓
\[\mathsf{hypot}\left(1, y\right) \cdot \left(\mathsf{hypot}\left(1, y\right) \cdot x\right) \]
x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{hypot}\left(1, y\right) \cdot \left(\mathsf{hypot}\left(1, y\right) \cdot x\right)

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))
↓
(FPCore (x y) :precision binary64 (* (hypot 1.0 y) (* (hypot 1.0 y) x)))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}
↓
double code(double x, double y) {
	return hypot(1.0, y) * (hypot(1.0, y) * x);
}

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. Simplified5.3

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

  3. Applied egg-rr6.5

    \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, y, 1\right)}\right)}^{3}} \]

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022130 
(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))))