Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.6% → 98.9%

Time: 3.8s

Alternatives: 5

Speedup: 1.9×

• 44.1% 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: 93.6% 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: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
3. Step-by-step derivation

   1. unpow2 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-17}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \left(x + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-17)
   (* x x)
   (if (<= x -7.5e-58)
     (* y y)
     (if (<= x -1.4e-80) (* x x) (* y (+ x (+ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-17) {
		tmp = x * x;
	} else if (x <= -7.5e-58) {
		tmp = y * y;
	} else if (x <= -1.4e-80) {
		tmp = x * x;
	} else {
		tmp = y * (x + (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.5d-17)) then
        tmp = x * x
    else if (x <= (-7.5d-58)) then
        tmp = y * y
    else if (x <= (-1.4d-80)) then
        tmp = x * x
    else
        tmp = y * (x + (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-17) {
		tmp = x * x;
	} else if (x <= -7.5e-58) {
		tmp = y * y;
	} else if (x <= -1.4e-80) {
		tmp = x * x;
	} else {
		tmp = y * (x + (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.5e-17:
		tmp = x * x
	elif x <= -7.5e-58:
		tmp = y * y
	elif x <= -1.4e-80:
		tmp = x * x
	else:
		tmp = y * (x + (x + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-17)
		tmp = Float64(x * x);
	elseif (x <= -7.5e-58)
		tmp = Float64(y * y);
	elseif (x <= -1.4e-80)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(x + Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.5e-17)
		tmp = x * x;
	elseif (x <= -7.5e-58)
		tmp = y * y;
	elseif (x <= -1.4e-80)
		tmp = x * x;
	else
		tmp = y * (x + (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.5e-17], N[(x * x), $MachinePrecision], If[LessEqual[x, -7.5e-58], N[(y * y), $MachinePrecision], If[LessEqual[x, -1.4e-80], N[(x * x), $MachinePrecision], N[(y * N[(x + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999978e-17 or -7.50000000000000002e-58 < x < -1.39999999999999995e-80

   1. Initial program 90.2%

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

      1. associate-+l+ 90.2%

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

      2. fma-def 90.2%

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

      3. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

      1. fma-udef 95.1%

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

      2. distribute-rgt-in 90.2%

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

      3. associate-+l+ 90.2%

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

      4. +-commutative 90.2%

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

      5. associate-*l* 90.2%

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

      6. distribute-lft-out 95.1%

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

   5. Applied egg-rr 95.1%

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

   6. Taylor expanded in y around 0 85.0%

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

      1. unpow2 85.0%

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

   8. Simplified 85.0%

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

   if -4.49999999999999978e-17 < x < -7.50000000000000002e-58

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.3%

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

      1. unpow2 74.3%

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

   4. Simplified 74.3%

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

   if -1.39999999999999995e-80 < x

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 66.3%

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

      1. unpow2 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-*l* 66.3%

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

      4. distribute-lft-in 68.7%

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

      5. fma-udef 68.7%

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

   4. Simplified 68.7%

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

      1. fma-udef 68.7%

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

      2. distribute-rgt-in 66.3%

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

      3. associate-*r* 66.3%

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

      4. *-commutative 66.3%

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

      5. *-commutative 66.3%

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

      6. associate-*r* 66.3%

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

      7. count-2 66.3%

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

      8. +-commutative 66.3%

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

      9. distribute-lft-in 66.3%

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

      10. associate-+r+ 66.3%

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

   6. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{\left(y \cdot y + y \cdot x\right) + y \cdot x} \]
   7. Step-by-step derivation

      1. distribute-lft-out 66.3%

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

      2. distribute-lft-out 68.7%

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

      3. +-commutative 68.7%

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

   8. Applied egg-rr 68.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-17}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \left(x + y\right)\right)\\ \end{array} \]

Alternative 3: 69.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-20} \lor \neg \left(x \leq -3.1 \cdot 10^{-58}\right) \land x \leq -3 \cdot 10^{-85}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.05e-20) (and (not (<= x -3.1e-58)) (<= x -3e-85)))
   (* x x)
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.05e-20) || (!(x <= -3.1e-58) && (x <= -3e-85))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.05d-20)) .or. (.not. (x <= (-3.1d-58))) .and. (x <= (-3d-85))) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.05e-20) || (!(x <= -3.1e-58) && (x <= -3e-85))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.05e-20) or (not (x <= -3.1e-58) and (x <= -3e-85)):
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.05e-20) || (!(x <= -3.1e-58) && (x <= -3e-85)))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.05e-20) || (~((x <= -3.1e-58)) && (x <= -3e-85)))
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.05e-20], And[N[Not[LessEqual[x, -3.1e-58]], $MachinePrecision], LessEqual[x, -3e-85]]], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0499999999999999e-20 or -3.0999999999999999e-58 < x < -3.00000000000000022e-85

   1. Initial program 90.5%

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

      1. associate-+l+ 90.5%

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

      2. fma-def 90.5%

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

      3. distribute-rgt-out 95.2%

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

   3. Simplified 95.2%

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

      1. fma-udef 95.2%

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

      2. distribute-rgt-in 90.5%

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

      3. associate-+l+ 90.5%

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

      4. +-commutative 90.5%

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

      5. associate-*l* 90.5%

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

      6. distribute-lft-out 95.2%

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

   5. Applied egg-rr 95.2%

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

   6. Taylor expanded in y around 0 83.2%

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

      1. unpow2 83.2%

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

   8. Simplified 83.2%

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

   if -1.0499999999999999e-20 < x < -3.0999999999999999e-58 or -3.00000000000000022e-85 < x

   1. Initial program 95.3%

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

   2. Taylor expanded in x around 0 68.0%

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

      1. unpow2 68.0%

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

   4. Simplified 68.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-20} \lor \neg \left(x \leq -3.1 \cdot 10^{-58}\right) \land x \leq -3 \cdot 10^{-85}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 69.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(x + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e-35) (* x (+ x (+ y y))) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-35) {
		tmp = x * (x + (y + y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d-35) then
        tmp = x * (x + (y + y))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-35) {
		tmp = x * (x + (y + y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e-35:
		tmp = x * (x + (y + y))
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e-35)
		tmp = Float64(x * Float64(x + Float64(y + y)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e-35)
		tmp = x * (x + (y + y));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e-35], N[(x * N[(x + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1500000000000001e-35

   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+ 95.5%

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

      2. fma-def 95.5%

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

      3. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

      1. fma-udef 97.7%

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

      2. distribute-rgt-in 95.5%

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

      3. associate-+l+ 95.5%

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

      4. +-commutative 95.5%

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

      5. associate-*l* 95.5%

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

      6. distribute-lft-out 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in y around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-*r* 61.9%

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

      3. count-2 61.9%

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

      4. distribute-rgt-in 64.2%

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

   8. Simplified 64.2%

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

   if 2.1500000000000001e-35 < y

   1. Initial program 89.9%

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

   2. Taylor expanded in x around 0 73.8%

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

      1. unpow2 73.8%

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

   4. Simplified 73.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(x + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 56.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-+l+ 93.7%

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

   2. fma-def 93.7%

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

   3. distribute-rgt-out 96.9%

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

3. Simplified 96.9%

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

   1. fma-udef 96.9%

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

   2. distribute-rgt-in 93.7%

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

   3. associate-+l+ 93.7%

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

   4. +-commutative 93.7%

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

   5. associate-*l* 93.7%

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

   6. distribute-lft-out 96.9%

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

5. Applied egg-rr 96.9%

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

6. Taylor expanded in y around 0 56.9%

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

   1. unpow2 56.9%

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

8. Simplified 56.9%

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

9. Final simplification 56.9%

   \[\leadsto x \cdot x \]

Developer target: 93.6% 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 2023195 
(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)))