Average Error: 5.3 → 0.1
Time: 2.8s
Precision: binary64
\[x \cdot \left(1 + y \cdot y\right) \]
↓
\[x + y \cdot \left(y \cdot x\right) \]
x \cdot \left(1 + y \cdot y\right)
↓
x + y \cdot \left(y \cdot x\right)

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))
↓
(FPCore (x y) :precision binary64 (+ x (* y (* y x))))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}
↓
double code(double x, double y) {
	return x + (y * (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. Taylor expanded in y around 0 5.3

    \[\leadsto \color{blue}{{y}^{2} \cdot x + x} \]

  4. Applied unpow2_binary645.3

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

  5. Applied associate-*l*_binary640.1

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

  6. Final simplification0.1

    \[\leadsto x + y \cdot \left(y \cdot x\right) \]

Reproduce

herbie shell --seed 2022067 
(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))))