Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.1% → 100.0%

Time: 6.5s

Alternatives: 5

Speedup: 1.9×

• 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: 94.1% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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. distribute-lft-out 97.2%

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

   3. fma-def 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-def 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. Final simplification 100.0%

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

Alternative 2: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-22) (* x x) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-22) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d-22)) then
        tmp = x * x
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-22) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-22:
		tmp = x * x
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-22)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-22)
		tmp = x * x;
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-22], N[(x * x), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004e-22

   1. Initial program 90.6%

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

   2. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   3. Step-by-step derivation

      1. unpow2 98.1%

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

   4. Simplified 98.1%

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

   5. Taylor expanded in x around inf 78.5%

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

      1. unpow2 78.5%

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

   7. Simplified 78.5%

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

   if -1.05000000000000004e-22 < x

   1. Initial program 95.0%

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

   2. Taylor expanded in x around 0 63.8%

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

   3. Taylor expanded in y around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. associate-*r* 63.8%

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

      5. distribute-lft-out 65.4%

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

   5. Simplified 65.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-22}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ y \cdot y + x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
3. Step-by-step derivation

   1. unpow2 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

   \[\leadsto y \cdot y + x \cdot x \]

Alternative 4: 68.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.3e-52) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-52) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.3d-52) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-52) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e-52:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-52)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-52)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-52], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999994e-52

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   3. Step-by-step derivation

      1. unpow2 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 66.9%

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

      1. unpow2 66.9%

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

   7. Simplified 66.9%

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

   if 2.29999999999999994e-52 < y

   1. Initial program 91.2%

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

   2. Taylor expanded in x around 0 72.5%

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

      1. unpow2 72.5%

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

   4. Simplified 72.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-52}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 5: 57.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
3. Step-by-step derivation

   1. unpow2 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 58.1%

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

   1. unpow2 58.1%

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

7. Simplified 58.1%

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

8. Final simplification 58.1%

   \[\leadsto x \cdot x \]

Developer target: 94.1% 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 2023274 
(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)))