Average Error: 0.0 → 0.0
Time: 1.8s
Precision: binary64
\[x + y \cdot \left(z + x\right) \]
↓
\[\mathsf{fma}\left(y, x, x + y \cdot z\right) \]
(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))
↓
(FPCore (x y z) :precision binary64 (fma y x (+ x (* y z))))
double code(double x, double y, double z) {
	return x + (y * (z + x));
}
↓
double code(double x, double y, double z) {
	return fma(y, x, (x + (y * z)));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Applied fma-def_binary640.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))