Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.5% → 100.0%

Time: 4.3s

Alternatives: 3

Speedup: 2.3×

• 44.6% 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: 93.5% 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, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. unpow2 N/A

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

   7. associate-+l+ N/A

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

   8. associate-+r+ N/A

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

   9. unpow2 N/A

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

   10. distribute-rgt-out N/A

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

   11. unpow2 N/A

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

   12. distribute-lft-out N/A

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

   13. distribute-rgt-out N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower-+.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(y + x\right) \cdot \left(y + x\right)} \]
6. Add Preprocessing

Alternative 2: 67.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-142}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.9e-142) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.9e-142) {
		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 <= (-1.9d-142)) 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 <= -1.9e-142) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.9e-142:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.9e-142)
		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 <= -1.9e-142)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.9e-142], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.89999999999999986e-142

   1. Initial program 94.2%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.9

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

   8. Applied rewrites 70.9%

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

   if -1.89999999999999986e-142 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 3: 55.9% accurate, 4.5× 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 53.7

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

8. Applied rewrites 53.7%

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

Developer Target 1: 93.5% 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 2024326 
(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)))