Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1

Percentage Accurate: 98.5% → 99.0%

Time: 3.6s

Precision: binary64

Cost: 7108

• Unsound rule application detected in e-graph. Results may not be sound.

• 33.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+272) (fma x x (* z (* y -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) {
	double tmp;
	if ((x * x) <= 1e+272) {
		tmp = fma(x, x, (z * (y * -4.0)));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * z))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+272)
		tmp = fma(x, x, Float64(z * Float64(y * -4.0)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+272], N[(x * x + N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.0000000000000001e272

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0%	\[ x \cdot x - \left(y \cdot 4\right) \cdot z \]
      fma-neg [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)} \]
      *-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x, x, -\color{blue}{z \cdot \left(y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 100.0%	\[ \mathsf{fma}\left(x, x, \color{blue}{z \cdot \left(-y \cdot 4\right)}\right) \]
      distribute-rgt-neg-in [=>] 100.0%	\[ \mathsf{fma}\left(x, x, z \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(x, x, z \cdot \left(y \cdot \color{blue}{-4}\right)\right) \]

   if 1.0000000000000001e272 < (*.f64 x x)

   1. Initial program 91.1%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {x}^{2} \]
      unpow2 [=>] 100.0%	\[ \color{blue}{x \cdot x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+272}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(y \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	6848

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

Alternative 3
Accuracy	99.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{+276}:\\ \;\;\;\;x \cdot x - z \cdot \left(y \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Accuracy	85.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 30000000000:\\ \;\;\;\;-4 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 8400000:\\ \;\;\;\;y \cdot \left(z \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Accuracy	52.7%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4.0) z)))