Average Error: 5.5 → 0.1
Time: 1.6s
Precision: binary64
\[x \cdot \left(1 + y \cdot y\right) \]
↓
\[\mathsf{fma}\left(y, y \cdot x, x\right) \]
x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{fma}\left(y, y \cdot x, x\right)

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

Error

Bits error versus x

Bits error versus y

Target

Original5.5
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y \]

Derivation

  1. Initial program 5.5

    \[x \cdot \left(1 + y \cdot y\right) \]

  2. Simplified5.5

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

  3. Taylor expanded in y around 0 5.5

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

  4. Applied unpow2_binary645.5

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

  5. Applied associate-*l*_binary640.1

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

  6. Applied fma-def_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022081 
(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))))