Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.6% → 97.2%

Time: 3.6s

Precision: binary64

Cost: 6916

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[x \cdot x - y \cdot y \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e+170) (fma x x (* y (- y))) (* x x)))

C

double code(double x, double y) {
	return (x * x) - (y * y);
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+170) {
		tmp = fma(x, x, (y * -y));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e+170)
		tmp = fma(x, x, Float64(y * Float64(-y)));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[x, 3.2e+170], N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999979e170

   1. Initial program 97.0%

      \[x \cdot x - y \cdot y \]

   2. Simplified 99.6%

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

      [Start] 97.0%	\[ x \cdot x - y \cdot y \]
      fma-neg [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(x, x, -y \cdot y\right)} \]
      distribute-rgt-neg-in [=>] 99.6%	\[ \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y\right)}\right) \]

   if 3.19999999999999979e170 < x

   1. Initial program 70.8%

      \[x \cdot x - y \cdot y \]

   2. Taylor expanded in x around inf 95.8%

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

   3. Simplified 95.8%

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

      [Start] 95.8%	\[ {x}^{2} \]
      unpow2 [=>] 95.8%	\[ \color{blue}{x \cdot x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 2
Accuracy	93.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+204}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]

Alternative 3
Accuracy	76.6%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-64} \lor \neg \left(y \leq 1.8 \cdot 10^{+65}\right):\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4
Accuracy	54.0%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))