Average Error: 0.0 → 0.1
Time: 1.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -\left(4 \cdot z\right) \cdot y\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -\left(4 \cdot z\right) \cdot y\right)
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(x, x, -((4.0 * z) * y));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x \cdot x - \left(y \cdot 4\right) \cdot z\]
  2. Using strategy rm

  3. Applied prod-diff0.0

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

  4. Simplified0.1

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

  5. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, x, -\left(4 \cdot z\right) \cdot y\right) + \color{blue}{0}\]

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))