Average Error: 0.0 → 0.1
Time: 2.0s
Precision: binary64
\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]
\[x \cdot x - \left(y \cdot 4\right) \cdot z \]
↓
\[\mathsf{fma}\left(y, -4 \cdot z, x \cdot x\right) \]
(FPCore (x y z) :precision binary64 (- (* x x) (* (* y 4.0) z)))
↓
(FPCore (x y z) :precision binary64 (fma y (* -4.0 z) (* x x)))
double code(double x, double y, double z) {
	return (x * x) - ((y * 4.0) * z);
}
↓
double code(double x, double y, double z) {
	return fma(y, (-4.0 * z), (x * x));
}

function code(x, y, z)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * z))
end
↓
function code(x, y, z)
	return fma(y, Float64(-4.0 * z), Float64(x * x))
end

code[x_, y_, z_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(y * N[(-4.0 * z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(y, -4 \cdot z, x \cdot x\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x \cdot x - \left(y \cdot 4\right) \cdot z \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(z \cdot -4\right)\right)} \]

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + -4 \cdot \left(y \cdot z\right)} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, -4 \cdot z, x \cdot x\right)} \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, -4 \cdot z, x \cdot x\right) \]

Reproduce

herbie shell --seed 2022170 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))