Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Average Error: 0.08% → 0.01%

Time: 2.3s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

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

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

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(x, x, Float64(Float64(y * z) * -4.0))
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[(x * x + N[(N[(y * z), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.08

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

2. Simplified 0.01

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

   [Start] 0.08	\[ x \cdot x - \left(y \cdot 4\right) \cdot z \]
   fma-neg [=>] 0.07	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)} \]
   *-commutative [=>] 0.07	\[ \mathsf{fma}\left(x, x, -\color{blue}{\left(4 \cdot y\right)} \cdot z\right) \]
   associate-*l* [=>] 0.01	\[ \mathsf{fma}\left(x, x, -\color{blue}{4 \cdot \left(y \cdot z\right)}\right) \]
   *-commutative [=>] 0.01	\[ \mathsf{fma}\left(x, x, -\color{blue}{\left(y \cdot z\right) \cdot 4}\right) \]
   distribute-rgt-neg-in [=>] 0.01	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot z\right) \cdot \left(-4\right)}\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(x, x, \left(y \cdot z\right) \cdot \color{blue}{-4}\right) \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	17.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+27}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-52}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.08%
Cost	576

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

Alternative 3
Error	56.01%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))