Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.8% → 99.9%

Time: 7.8s

Alternatives: 5

Speedup: 1.9×

• 43.1% 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: 99.9% 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 95.3%

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

   1. associate-*l* 95.3%

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

   2. *-commutative 95.3%

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

   3. distribute-lft-out 98.0%

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

   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. Add Preprocessing

Alternative 2: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-105) (* x x) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-105) {
		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 <= (-6.8d-105)) 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 <= -6.8e-105) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-105:
		tmp = x * x
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e-105)
		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 <= -6.8e-105)
		tmp = x * x;
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-105], 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 < -6.79999999999999984e-105

   1. Initial program 94.9%

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

      1. associate-+l+ 94.9%

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

      2. +-commutative 94.9%

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

      3. fma-define 94.9%

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

      4. associate-*l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*r* 79.4%

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

      3. *-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in x around 0 81.4%

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

   9. Taylor expanded in x around inf 80.2%

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

   if -6.79999999999999984e-105 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 98.7%

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

   4. Taylor expanded in x around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. distribute-rgt-out 66.6%

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

      4. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-105}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5.1e-105) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.1e-105) {
		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 (x <= (-5.1d-105)) 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 (x <= -5.1e-105) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.1e-105:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.1e-105)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.1e-105)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.1e-105], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.10000000000000007e-105

   1. Initial program 94.9%

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

      1. associate-+l+ 94.9%

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

      2. +-commutative 94.9%

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

      3. fma-define 94.9%

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

      4. associate-*l* 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*r* 79.4%

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

      3. *-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in x around 0 81.4%

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

   9. Taylor expanded in x around inf 80.2%

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

   if -5.10000000000000007e-105 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 98.7%

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

   4. Taylor expanded in x around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

      3. distribute-rgt-out 66.6%

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

      4. *-commutative 66.6%

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

   6. Applied egg-rr 66.6%

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

   7. Taylor expanded in x around 0 65.1%

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

Alternative 4: 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 95.3%

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

   1. associate-+l+ 95.3%

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

   2. +-commutative 95.3%

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

   3. fma-define 95.3%

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

   4. associate-*l* 95.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in y around 0 97.2%

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

6. Taylor expanded in y around inf 99.3%

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

7. Final simplification 99.3%

   \[\leadsto y \cdot y + x \cdot x \]
8. Add Preprocessing

Alternative 5: 57.2% 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 95.3%

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

   1. associate-+l+ 95.3%

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

   2. +-commutative 95.3%

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

   3. fma-define 95.3%

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

   4. associate-*l* 95.3%

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

3. Simplified 95.3%

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

5. Taylor expanded in y around 0 56.6%

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

   1. *-commutative 56.6%

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

   2. associate-*r* 56.6%

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

   3. *-commutative 56.6%

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

7. Simplified 56.6%

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

8. Taylor expanded in x around 0 59.3%

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

9. Taylor expanded in x around inf 58.4%

   \[\leadsto x \cdot \color{blue}{x} \]
10. Add Preprocessing

Developer Target 1: 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 2024170 
(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)))