Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.8% → 99.4%

Time: 3.1s

Alternatives: 5

Speedup: 1.2×

• 43.7% 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: 93.8% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e+132) (+ (* x (+ x (* 2.0 y))) (* y y)) (pow y 2.0)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+132) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = pow(y, 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 (y <= 3.8d+132) then
        tmp = (x * (x + (2.0d0 * y))) + (y * y)
    else
        tmp = y ** 2.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+132) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = Math.pow(y, 2.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e+132:
		tmp = (x * (x + (2.0 * y))) + (y * y)
	else:
		tmp = math.pow(y, 2.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e+132)
		tmp = Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y));
	else
		tmp = y ^ 2.0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e+132)
		tmp = (x * (x + (2.0 * y))) + (y * y);
	else
		tmp = y ^ 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[y, 3.8e+132], N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[Power[y, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000006e132

   1. Initial program 95.3%

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

      1. +-commutative 95.3%

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

      2. associate-*l* 95.3%

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

      3. distribute-lft-out 98.6%

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

   4. Applied egg-rr 98.6%

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

   if 3.80000000000000006e132 < y

   1. Initial program 81.0%

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

   3. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+221}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+221)
   (+ (* x (+ x (* 2.0 y))) (* y y))
   (fma x x (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+221) {
		tmp = (x * (x + (2.0 * y))) + (y * y);
	} else {
		tmp = fma(x, x, (y * (y + (x * 2.0))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+221)
		tmp = Float64(Float64(x * Float64(x + Float64(2.0 * y))) + Float64(y * y));
	else
		tmp = fma(x, x, Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.5e+221], N[(N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000035e221

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-*l* 75.0%

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

      3. distribute-lft-out 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -7.50000000000000035e221 < x

   1. Initial program 94.2%

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

      1. associate-+l+ 94.2%

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

      2. associate-*l* 94.2%

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

      3. *-commutative 94.2%

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

      4. *-commutative 94.2%

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

      5. +-commutative 94.2%

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

      6. fma-define 94.2%

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

      7. *-commutative 94.2%

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

      8. *-commutative 94.2%

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

      9. associate-*l* 94.2%

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

      10. distribute-rgt-out 98.3%

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

      11. +-commutative 98.3%

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

   3. Simplified 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+221}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (x * (x + (2.0 * y))) + (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 = (x * (x + (2.0d0 * y))) + (y * y)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x * (x + (2.0 * y))) + (y * y)

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x * (x + (2.0 * y))) + (y * y);
end

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. +-commutative 93.0%

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

   2. associate-*l* 93.0%

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

   3. distribute-lft-out 96.1%

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

4. Applied egg-rr 96.1%

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

5. Final simplification 96.1%

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

Alternative 4: 58.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 * 2.0d0))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * (y + (x * 2.0))

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * (y + (x * 2.0));
end

Wolfram

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

Derivation

1. Initial program 93.0%

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

3. Taylor expanded in x around 0 52.0%

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

   1. associate-*r* 52.0%

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

   2. distribute-rgt-out 55.9%

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

5. Applied egg-rr 55.9%

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

6. Final simplification 55.9%

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

Alternative 5: 15.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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 * (x * 2.0d0)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * (x * 2.0)

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * (x * 2.0);
end

Wolfram

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

Derivation

1. Initial program 93.0%

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

3. Taylor expanded in x around 0 52.0%

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

4. Taylor expanded in x around inf 14.7%

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

   1. associate-*r* 14.7%

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

   2. *-commutative 14.7%

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

6. Simplified 14.7%

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

7. Final simplification 14.7%

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

Developer target: 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 2024105 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

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