Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.0% → 99.5%

Time: 3.3s

Alternatives: 4

Speedup: 0.8×

• 43.1% 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: 94.0% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.9e+128) (+ (* x (+ x (* y 2.0))) (* y y)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+128) {
		tmp = (x * (x + (y * 2.0))) + (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 <= 2.9d+128) then
        tmp = (x * (x + (y * 2.0d0))) + (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 <= 2.9e+128) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.9e+128:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.9e+128)
		tmp = Float64(Float64(x * Float64(x + Float64(y * 2.0))) + 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 <= 2.9e+128)
		tmp = (x * (x + (y * 2.0))) + (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, 2.9e+128], N[(N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e128

   1. Initial program 95.4%

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

      1. +-commutative 95.4%

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

      2. associate-*l* 95.4%

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

      3. distribute-lft-out 96.8%

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

   4. Applied egg-rr 96.8%

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

   if 2.9e128 < y

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. pow2 97.7%

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

   5. Applied egg-rr 97.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-298}:\\ \;\;\;\;y \cdot y + y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-298) (+ (* y y) (* y (* x 2.0))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-298) {
		tmp = (y * y) + (y * (x * 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 <= (-3.3d-298)) then
        tmp = (y * y) + (y * (x * 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 <= -3.3e-298) {
		tmp = (y * y) + (y * (x * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.3e-298:
		tmp = (y * y) + (y * (x * 2.0))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-298)
		tmp = Float64(Float64(y * y) + Float64(y * Float64(x * 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 <= -3.3e-298)
		tmp = (y * y) + (y * (x * 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, -3.3e-298], N[(N[(y * y), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3000000000000002e-298

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*r* 55.7%

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

      3. *-commutative 55.7%

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

      4. associate-*r* 55.7%

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

   5. Simplified 55.7%

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

   if -3.3000000000000002e-298 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0 53.9%

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

      1. pow2 53.9%

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

   5. Applied egg-rr 53.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-298}:\\ \;\;\;\;y \cdot y + y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-298}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.3e-298) (* y (* x 2.0)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.3e-298) {
		tmp = y * (x * 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 <= (-3.3d-298)) then
        tmp = y * (x * 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 <= -3.3e-298) {
		tmp = y * (x * 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.3e-298:
		tmp = y * (x * 2.0)
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.3e-298)
		tmp = Float64(y * Float64(x * 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 <= -3.3e-298)
		tmp = y * (x * 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, -3.3e-298], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3000000000000002e-298

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*r* 55.7%

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

      3. *-commutative 55.7%

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

      4. associate-*r* 55.7%

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

   5. Simplified 55.7%

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

   6. Taylor expanded in y around 0 15.3%

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

      1. *-commutative 15.3%

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

      2. associate-*r* 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.3%

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

   8. Simplified 15.3%

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

   if -3.3000000000000002e-298 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0 53.9%

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

      1. pow2 53.9%

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

   5. Applied egg-rr 53.9%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-298}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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 = y * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in x around 0 56.3%

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

   1. pow2 56.3%

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

5. Applied egg-rr 56.3%

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

6. Final simplification 56.3%

   \[\leadsto y \cdot y \]
7. Add Preprocessing

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 2024036 
(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)))