Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.6% → 99.9%

Time: 4.7s

Precision: binary64

Cost: 7876

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

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+252)
   (fma x x (* y (+ (* x 2.0) y)))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

↓

double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+252) {
		tmp = fma(x, x, (y * ((x * 2.0) + y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

↓

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+252)
		tmp = fma(x, x, Float64(y * Float64(Float64(x * 2.0) + y)));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+252], N[(x * x + N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	93.6%
Target	93.6%
Herbie	99.9%

\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 4.9999999999999997e252

   1. Initial program 99.9%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

   2. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
      associate-+l+ [=>] 99.9%	\[ \color{blue}{x \cdot x + \left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]
      fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]
      distribute-rgt-out [=>] 99.9%	\[ \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(x \cdot 2 + y\right)}\right) \]

   if 4.9999999999999997e252 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 89.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]

   3. Simplified 100.0%

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

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

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+252}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 3
Accuracy	67.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \left(x + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	448

\[x \cdot x + y \cdot y \]

Alternative 5
Accuracy	67.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6
Accuracy	57.8%
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))