Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.2% → 98.8%
Time: 2.7s
Alternatives: 4
Speedup: 1.2×

Specification

?
\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

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

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

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

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

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

\begin{array}{l}

\\
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 4 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 93.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function

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

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

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

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

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

\begin{array}{l}

\\
\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
\end{array}

Alternative 1: 98.8% accurate, 0.1× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+234}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.35e+234)
   (+ (* x (+ x (* 2.0 y))) (* y y))
   (fma x x (* y (+ y (* x 2.0))))))
assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.35e+234) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = fma(x, x, (y * (y + (x * 2.0))));
	}
	return tmp;
}

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.35e+234)
		tmp = Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y));
	else
		tmp = fma(x, x, Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.35e+234], N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.35 \cdot 10^{+234}:\\
\;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.15 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.15e+139) (+ (* x (+ x (* 2.0 y))) (* y y)) (pow y 2.0)))
assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.15e+139) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = pow(y, 2.0);
	}
	return tmp;
}

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.15d+139) then
        tmp = (x * (x + (2.0d0 * y))) + (y * y)
    else
        tmp = y ** 2.0d0
    end if
    code = tmp
end function

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.15e+139) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = Math.pow(y, 2.0);
	}
	return tmp;
}

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.15e+139:
		tmp = (x * (x + (2.0 * y))) + (y * y)
	else:
		tmp = math.pow(y, 2.0)
	return tmp

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.15e+139)
		tmp = Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y));
	else
		tmp = y ^ 2.0;
	end
	return tmp
end

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.15e+139)
		tmp = (x * (x + (2.0 * y))) + (y * y);
	else
		tmp = y ^ 2.0;
	end
	tmp_2 = tmp;
end

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.15e+139], N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[Power[y, 2.0], $MachinePrecision]]

\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
\begin{array}{l}
\mathbf{if}\;y \leq 3.15 \cdot 10^{+139}:\\
\;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\

\mathbf{else}:\\
\;\;\;\;{y}^{2}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 96.7% accurate, 1.2× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot \left(x + 2 \cdot y\right) + y \cdot y \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ (* x (+ x (* 2.0 y))) (* y y)))
assert(x < y);
double code(double x, double y) {
	return (x * (x + (2.0 * y))) + (y * y);
}

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * (x + (2.0d0 * y))) + (y * y)
end function

assert x < y;
public static double code(double x, double y) {
	return (x * (x + (2.0 * y))) + (y * y);
}

[x, y] = sort([x, y])
def code(x, y):
	return (x * (x + (2.0 * y))) + (y * y)

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y))
end

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x * (x + (2.0 * y))) + (y * y);
end

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
x \cdot \left(x + 2 \cdot y\right) + y \cdot y
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 54.2% accurate, 1.4× speedup?

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot y + y \cdot \left(x \cdot 2\right) \end{array} \]
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (+ (* y y) (* y (* x 2.0))))
assert(x < y);
double code(double x, double y) {
	return (y * y) + (y * (x * 2.0));
}

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + (y * (x * 2.0d0))
end function

assert x < y;
public static double code(double x, double y) {
	return (y * y) + (y * (x * 2.0));
}

[x, y] = sort([x, y])
def code(x, y):
	return (y * y) + (y * (x * 2.0))

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y * y) + Float64(y * Float64(x * 2.0)))
end

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y * y) + (y * (x * 2.0));
end

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}
[x, y] = \mathsf{sort}([x, y])\\
\\
y \cdot y + y \cdot \left(x \cdot 2\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Developer target: 93.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \end{array} \]
(FPCore (x y) :precision binary64 (+ (* x x) (+ (* y y) (* (* x y) 2.0))))
double code(double x, double y) {
	return (x * x) + ((y * y) + ((x * y) * 2.0));
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) + ((x * y) * 2.0d0))
end function

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

def code(x, y):
	return (x * x) + ((y * y) + ((x * y) * 2.0))

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

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + ((x * y) * 2.0));
end

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

\begin{array}{l}

\\
x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right)
\end{array}

Reproduce

?
herbie shell --seed 2024008 
(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)))