Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Average Error: 0.0 → 0.0

Time: 3.6s

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
   (fma.f64 y (*.f64 z -4) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (*.f64 z (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z 4))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 z))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 (*.f64 4 z))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y (*.f64 4 z)))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y 4) z))) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) (neg.f64 (*.f64 (*.f64 y 4) z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) z))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	11.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -140:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 29000000000000:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Error	0.0
Cost	576

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

Alternative 3
Error	35.7
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022349 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))