Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Average Accuracy: 99.9% → 99.9%

Time: 3.6s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma x x (* z (* y -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, (z * (y * -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(z * Float64(y * -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[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ x \cdot x - \left(y \cdot 4\right) \cdot z \]
   fma-neg [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)} \]
   *-commutative [=>] 99.9	\[ \mathsf{fma}\left(x, x, -\color{blue}{z \cdot \left(y \cdot 4\right)}\right) \]
   distribute-rgt-neg-in [=>] 99.9	\[ \mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(-y \cdot 4\right)}\right) \]
   distribute-rgt-neg-in [=>] 99.9	\[ \mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
   metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(x, x, z \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	82.3%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -120000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Accuracy	82.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -115000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-49}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	576

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

Alternative 4
Accuracy	44.7%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))