Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.7% → 99.6%

Time: 3.4s

Alternatives: 5

Speedup: 0.8×

• 42.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+135}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9e+135) (+ (* y y) (* x (+ x (* y 2.0)))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9e+135) {
		tmp = (y * y) + (x * (x + (y * 2.0)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

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 <= 9d+135) then
        tmp = (y * y) + (x * (x + (y * 2.0d0)))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9e+135) {
		tmp = (y * y) + (x * (x + (y * 2.0)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9e+135:
		tmp = (y * y) + (x * (x + (y * 2.0)))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9e+135)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(y * 2.0))));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e+135)
		tmp = (y * y) + (x * (x + (y * 2.0)));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9e+135], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000014e135

   1. Initial program 95.5%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 97.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto x \cdot x + \left(\left(x \cdot 2\right) \cdot y + \color{blue}{y \cdot y}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + \color{blue}{y \cdot y} \]

      3. +-commutative N/A

         \[\leadsto y \cdot y + \color{blue}{\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y \cdot y\right), \color{blue}{\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{x \cdot x} + \left(x \cdot 2\right) \cdot y\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot x + x \cdot \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{\left(x + 2 \cdot y\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(x + y \cdot \color{blue}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x + y \cdot 2\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot 2\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right)\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{y \cdot y + x \cdot \left(x + y \cdot 2\right)} \]

   if 9.00000000000000014e135 < y

   1. Initial program 88.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 94.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{y}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{y} \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+172) (* x x) (+ (* x x) (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+172) {
		tmp = x * x;
	} else {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	}
	return tmp;
}

Fortran

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 (x <= (-5d+172)) then
        tmp = x * x
    else
        tmp = (x * x) + (y * (y + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+172) {
		tmp = x * x;
	} else {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+172:
		tmp = x * x
	else:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+172)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(x * x) + Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+172)
		tmp = x * x;
	else
		tmp = (x * x) + (y * (y + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+172], N[(x * x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000001e172

   1. Initial program 81.8%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 81.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x} \]

   if -5.0000000000000001e172 < x

   1. Initial program 95.7%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 98.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+172}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-112}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-112) (* x (+ x (* y 2.0))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-112) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

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 <= 7d-112) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-112) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7e-112:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7e-112)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-112)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7e-112], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999988e-112

   1. Initial program 95.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{1} + 2 \cdot \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\left(1 + 2 \cdot \frac{y}{x}\right) \cdot \color{blue}{x}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(1 + 2 \cdot \frac{y}{x}\right) \cdot x\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(2 \cdot \frac{y}{x} + 1\right) \cdot x\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x + \color{blue}{\left(2 \cdot \frac{y}{x}\right) \cdot x}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x + \frac{2 \cdot y}{x} \cdot x\right)\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x + \frac{\left(2 \cdot y\right) \cdot x}{\color{blue}{x}}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x + \left(2 \cdot y\right) \cdot \color{blue}{\frac{x}{x}}\right)\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x + \left(2 \cdot y\right) \cdot 1\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(x + 2 \cdot \color{blue}{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

   7. Simplified 66.5%

      \[\leadsto \color{blue}{x \cdot \left(x + y \cdot 2\right)} \]

   if 6.99999999999999988e-112 < y

   1. Initial program 93.3%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 95.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 95.5%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{y}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{y} \]

      2. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 78.4%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-112}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 6.4e-112) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-112) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

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 <= 6.4d-112) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-112) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.4e-112:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.4e-112)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e-112)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.4e-112], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.39999999999999986e-112

   1. Initial program 95.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x \cdot \color{blue}{x} \]

      2. *-lowering-*.f64 66.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 66.1%

      \[\leadsto \color{blue}{x \cdot x} \]

   if 6.39999999999999986e-112 < y

   1. Initial program 93.3%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 95.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 95.5%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{y}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{y} \]

      2. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 78.4%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 57.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x x))

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto x \cdot x + \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

   7. *-lowering-*.f64 96.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

3. Simplified 96.9%

   \[\leadsto \color{blue}{x \cdot x + y \cdot \left(x \cdot 2 + y\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{2}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto x \cdot \color{blue}{x} \]

   2. *-lowering-*.f64 53.4%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

7. Simplified 53.4%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Developer Target 1: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (+ (* y y) (* (* x y) 2))))

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