Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Average Error: 0.1 → 0.1

Time: 2.2s

Precision: binary64

Cost: 6848

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]

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 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right)} \]
   Proof
   (fma.f64 x x (*.f64 z (*.f64 y -4))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (*.f64 z (*.f64 y (Rewrite<= metadata-eval (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (*.f64 z (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y 4))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z (*.f64 y 4))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y 4) z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) z))): 3 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	11.0
Cost	1100

\[\begin{array}{l} t_0 := -4 \cdot \left(z \cdot y\right)\\ \mathbf{if}\;x \cdot x \leq 2.35 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 7.5 \cdot 10^{-150}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 6600000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	576

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

Alternative 3
Error	35.9
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))