Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.9% → 99.9%

Time: 2.8s

Alternatives: 5

Speedup: 1.0×

• 44.0% 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: 93.9% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+151}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e+151) (+ (* y y) (* x (+ x (* y 2.0)))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+151) {
		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 <= 2.1d+151) 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 <= 2.1e+151) {
		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 <= 2.1e+151:
		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 <= 2.1e+151)
		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 <= 2.1e+151)
		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, 2.1e+151], 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 < 2.1000000000000001e151

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. fma-def 97.4%

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

      3. distribute-rgt-out 97.8%

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

   3. Simplified 97.8%

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

      1. fma-udef 97.8%

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

      2. distribute-rgt-in 97.4%

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

      3. associate-+l+ 97.4%

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

      4. +-commutative 97.4%

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

      5. associate-*l* 97.4%

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

      6. distribute-lft-out 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 2.1000000000000001e151 < y

   1. Initial program 84.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+151}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma x x (* y (+ y (* x 2.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. associate-+l+ 96.1%

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

   2. fma-def 96.1%

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

   3. distribute-rgt-out 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 3: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(x + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-32) (* x (+ x (+ y y))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-32) {
		tmp = x * (x + (y + y));
	} 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 <= 3.2d-32) then
        tmp = x * (x + (y + y))
    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 <= 3.2e-32) {
		tmp = x * (x + (y + y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-32:
		tmp = x * (x + (y + y))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-32)
		tmp = Float64(x * Float64(x + Float64(y + y)));
	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 <= 3.2e-32)
		tmp = x * (x + (y + y));
	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, 3.2e-32], N[(x * N[(x + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000002e-32

   1. Initial program 98.0%

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

      1. associate-+l+ 98.0%

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

      2. fma-def 98.0%

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

      3. distribute-rgt-out 98.5%

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

   3. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. distribute-rgt-in 98.0%

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

      3. associate-+l+ 98.0%

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

      4. +-commutative 98.0%

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

      5. associate-*l* 98.0%

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

      6. distribute-lft-out 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in y around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. count-2 66.0%

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

      3. unpow2 66.0%

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

      4. distribute-rgt-in 67.5%

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

   8. Simplified 67.5%

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

   if 3.2000000000000002e-32 < y

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 81.2%

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

      1. unpow2 81.2%

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

   4. Simplified 81.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \left(x + \left(y + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 86.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-32}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.6e-32) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-32) {
		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 <= 1.6d-32) 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 <= 1.6e-32) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.6e-32:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-32)
		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 <= 1.6e-32)
		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, 1.6e-32], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6000000000000001e-32

   1. Initial program 98.0%

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

   2. Taylor expanded in x around inf 65.0%

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

      1. unpow2 65.0%

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

   4. Simplified 65.0%

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

   if 1.6000000000000001e-32 < y

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 81.2%

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

      1. unpow2 81.2%

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

   4. Simplified 81.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-32}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

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 96.1%

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

2. Taylor expanded in x around inf 57.0%

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

   1. unpow2 57.0%

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

4. Simplified 57.0%

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

5. Final simplification 57.0%

   \[\leadsto x \cdot x \]

Developer target: 94.0% 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 2023199 
(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)))