Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.3% → 99.0%

Time: 6.8s

Alternatives: 4

Speedup: 1.9×

• 44.2% 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: 93.3% 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: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-+l+ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

   5. +-commutative 91.0%

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

   6. fma-define 91.0%

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

   7. *-commutative 91.0%

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

   8. *-commutative 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in y around 0 96.5%

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

6. Taylor expanded in y around inf 98.8%

   \[\leadsto x \cdot x + y \cdot \color{blue}{y} \]
7. Add Preprocessing

Alternative 2: 36.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-305}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7e-305) (* x x) (* y (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e-305) {
		tmp = x * x;
	} else {
		tmp = 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 <= (-7d-305)) then
        tmp = x * x
    else
        tmp = y * (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7e-305) {
		tmp = x * x;
	} else {
		tmp = y * (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7e-305:
		tmp = x * x
	else:
		tmp = y * (x * 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e-305)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7e-305)
		tmp = x * x;
	else
		tmp = y * (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e-305], N[(x * x), $MachinePrecision], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999996e-305

   1. Initial program 93.1%

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

      1. associate-+l+ 93.1%

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

      2. associate-*l* 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

      5. +-commutative 93.1%

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

      6. fma-define 93.1%

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

      7. *-commutative 93.1%

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

      8. *-commutative 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*r* 53.9%

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

      3. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in x around 0 56.5%

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

   9. Taylor expanded in x around inf 55.8%

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

   if -6.9999999999999996e-305 < x

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. associate-*l* 89.3%

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

      3. *-commutative 89.3%

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

      4. *-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. fma-define 89.3%

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

      7. *-commutative 89.3%

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

      8. *-commutative 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*r* 53.7%

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

      3. *-commutative 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in x around 0 14.6%

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

      1. associate-*r* 14.6%

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

   10. Simplified 14.6%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-305}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * (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 * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-+l+ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

   5. +-commutative 91.0%

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

   6. fma-define 91.0%

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

   7. *-commutative 91.0%

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

   8. *-commutative 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in y around 0 53.8%

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

   1. *-commutative 53.8%

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

   2. associate-*r* 53.8%

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

   3. *-commutative 53.8%

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

7. Simplified 53.8%

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

8. Taylor expanded in x around 0 57.3%

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

9. Final simplification 57.3%

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

Alternative 4: 57.6% 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 91.0%

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

   1. associate-+l+ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

   5. +-commutative 91.0%

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

   6. fma-define 91.0%

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

   7. *-commutative 91.0%

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

   8. *-commutative 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in y around 0 53.8%

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

   1. *-commutative 53.8%

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

   2. associate-*r* 53.8%

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

   3. *-commutative 53.8%

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

7. Simplified 53.8%

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

8. Taylor expanded in x around 0 57.3%

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

9. Taylor expanded in x around inf 57.3%

   \[\leadsto x \cdot \color{blue}{x} \]
10. 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 2024136 
(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)))