Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.4% → 99.9%

Time: 4.8s

Alternatives: 5

Speedup: 1.9×

• 43.3% 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: 93.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

   1. associate-*l* 92.2%

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

   2. *-commutative 92.2%

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

   3. distribute-lft-out 95.7%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. fma-define 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-39) (* y (+ y (* x 2.0))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-39) {
		tmp = 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 <= 1.5d-39) then
        tmp = 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 <= 1.5e-39) {
		tmp = 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 <= 1.5e-39:
		tmp = 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 <= 1.5e-39)
		tmp = Float64(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 <= 1.5e-39)
		tmp = 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, 1.5e-39], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000014e-39

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0 96.1%

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

   4. Taylor expanded in x around 0 47.8%

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

      1. +-commutative 47.8%

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

      2. unpow2 47.8%

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

      3. associate-*r* 47.8%

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

      4. distribute-rgt-in 51.7%

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

   6. Simplified 51.7%

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

   if 1.50000000000000014e-39 < y

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0 94.6%

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

   4. Taylor expanded in x around 0 71.3%

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

      1. +-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. associate-*r* 71.3%

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

      4. distribute-rgt-in 76.7%

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

   6. Simplified 76.7%

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

   7. Taylor expanded in y around inf 78.2%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-39}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-298}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.1e-298) (* y (* x 2.0)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-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 <= 1.1d-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 <= 1.1e-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 <= 1.1e-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 <= 1.1e-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 <= 1.1e-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, 1.1e-298], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-298

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0 94.4%

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

   4. Taylor expanded in x around 0 54.6%

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

      1. +-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. associate-*r* 54.6%

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

      4. distribute-rgt-in 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in y around 0 17.2%

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

      1. associate-*r* 17.2%

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

      2. *-commutative 17.2%

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

   9. Simplified 17.2%

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

   if 1.1e-298 < y

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0 96.9%

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

   4. Taylor expanded in x around 0 54.6%

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

      1. +-commutative 54.6%

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

      2. unpow2 54.6%

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

      3. associate-*r* 54.6%

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

      4. distribute-rgt-in 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in y around inf 58.1%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-298}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (y * y) + (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 = (y * y) + (x * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in x around 0 95.7%

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

4. Taylor expanded in x around inf 99.4%

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

5. Final simplification 99.4%

   \[\leadsto y \cdot y + x \cdot x \]
6. Add Preprocessing

Alternative 5: 57.6% 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 92.2%

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

3. Taylor expanded in x around 0 95.7%

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

4. Taylor expanded in x around 0 54.6%

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

   1. +-commutative 54.6%

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

   2. unpow2 54.6%

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

   3. associate-*r* 54.6%

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

   4. distribute-rgt-in 58.9%

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

6. Simplified 58.9%

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

7. Taylor expanded in y around inf 59.0%

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

Developer Target 1: 93.4% 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 2024181 
(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)))