Average Error: 5.2 → 0.1
Time: 1.7s
Precision: binary64
\[x \cdot \left(1 + y \cdot y\right) \]
↓
\[\mathsf{fma}\left(y, y \cdot x, x\right) \]
(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);
}

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end
↓
function code(x, y)
	return fma(y, Float64(y * x), x)
end

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(y * N[(y * x), $MachinePrecision] + x), $MachinePrecision]

x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{fma}\left(y, y \cdot x, x\right)

Error

Bits error versus x

Bits error versus y

Target

Original5.2
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y \]

Derivation

  1. Initial program 5.2

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

  2. Simplified5.2

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

  3. Taylor expanded in x around 0 5.2

    \[\leadsto \color{blue}{\left(1 + {y}^{2}\right) \cdot x} \]

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022160 
(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))))