Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 99.9%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x + y)
end function

Java

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

Python

def code(x, y):
	return (x + y) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x + y)
end function

Java

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

Python

def code(x, y):
	return (x + y) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(y, fma(2.0, x, 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[(2.0 * x + y), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 95.3%

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

   1. +-commutative 95.3%

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

   2. associate-+r+ 95.3%

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

   3. associate-*r* 95.3%

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

   4. unpow2 95.3%

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

   5. distribute-rgt-out 97.6%

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

   6. fma-define 99.6%

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

   7. fma-define 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 59.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. distribute-lft-out 58.8%

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

5. Simplified 58.8%

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

6. Final simplification 58.8%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 4: 15.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-*l* 56.5%

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

   3. unpow2 56.5%

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

   4. distribute-lft-out 58.8%

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

5. Simplified 58.8%

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

6. Taylor expanded in x around 0 17.5%

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

7. Final simplification 17.5%

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

Developer target: 93.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (+ (* y y) (* 2.0 (* y x)))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) + (2.0d0 * (y * x)))
end function

Java

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

Python

def code(x, y):
	return (x * x) + ((y * y) + (2.0 * (y * x)))

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(2.0 * Float64(y * x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + (2.0 * (y * x)));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024039 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

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