Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.7% → 99.9%

Time: 4.0s

Alternatives: 5

Speedup: 0.8×

• 42.8% 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: 93.7% 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 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%

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

   2. *-commutative 93.7%

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

   3. distribute-lft-out 97.7%

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

   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. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+154}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2e+154) (+ (* y y) (* x (+ x (* 2.0 y)))) (pow y 2.0)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e+154) {
		tmp = (y * y) + (x * (x + (2.0 * 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 <= 2d+154) then
        tmp = (y * y) + (x * (x + (2.0d0 * 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 <= 2e+154) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = Math.pow(y, 2.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2e+154:
		tmp = (y * y) + (x * (x + (2.0 * 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 <= 2e+154)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * 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 <= 2e+154)
		tmp = (y * y) + (x * (x + (2.0 * 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, 2e+154], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[y, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000007e154

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 2.00000000000000007e154 < y

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+154}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+238}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e+238) (+ (* y y) (* x (+ x (* 2.0 y)))) (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.2e+238) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * (y + (x * 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 <= 9.2d+238) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e+238) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.2e+238:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.2e+238)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e+238)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = y * (y + (x * 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, 9.2e+238], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2000000000000001e238

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 9.2000000000000001e238 < y

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0 64.3%

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

      1. associate-*r* 64.3%

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

      2. *-commutative 64.3%

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

      3. *-commutative 64.3%

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

      4. *-commutative 64.3%

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

   5. Simplified 64.3%

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

   6. Taylor expanded in y around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+238}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% 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 93.7%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
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. associate-*r* 55.7%

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

   2. *-commutative 55.7%

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

   3. *-commutative 55.7%

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

   4. *-commutative 55.7%

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

5. Simplified 55.7%

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

6. Taylor expanded in y around 0 58.1%

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

7. Final simplification 58.1%

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

Alternative 5: 15.3% 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 93.7%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
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. associate-*r* 55.7%

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

   2. *-commutative 55.7%

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

   3. *-commutative 55.7%

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

   4. *-commutative 55.7%

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

5. Simplified 55.7%

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

6. Taylor expanded in y around 0 58.1%

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

7. Taylor expanded in y around 0 13.9%

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

   1. associate-*r* 13.9%

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

   2. *-commutative 13.9%

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

9. Simplified 13.9%

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

10. Final simplification 13.9%

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

Developer target: 93.7% 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 2024089 
(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)))