Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Average Error: 0.0 → 0.0

Time: 2.4s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x x) (* (* y 4.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma y (* z -4.0) (* x x)))

C

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, (z * -4.0), (x * x));
}

Julia

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(z * -4.0), Float64(x * x))
end

Wolfram

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[(z * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ x \cdot x - \left(y \cdot 4\right) \cdot z \]
   sub-neg [=>] 0.0	\[ \color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot z\right)} \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(-\left(y \cdot 4\right) \cdot z\right) + x \cdot x} \]
   associate-*l* [=>] 0.0	\[ \left(-\color{blue}{y \cdot \left(4 \cdot z\right)}\right) + x \cdot x \]
   distribute-rgt-neg-in [=>] 0.0	\[ \color{blue}{y \cdot \left(-4 \cdot z\right)} + x \cdot x \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(y, -4 \cdot z, x \cdot x\right)} \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(y, -\color{blue}{z \cdot 4}, x \cdot x\right) \]
   distribute-rgt-neg-in [=>] 0.0	\[ \mathsf{fma}\left(y, \color{blue}{z \cdot \left(-4\right)}, x \cdot x\right) \]
   metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(y, z \cdot \color{blue}{-4}, x \cdot x\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	11.1
Cost	1101

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.85 \cdot 10^{-31} \lor \neg \left(x \cdot x \leq 45000000000000\right) \land x \cdot x \leq 6.2 \cdot 10^{+28}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	576

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

Alternative 3
Error	36.2
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))