Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.2% → 99.3%

Time: 8.4s

Alternatives: 3

Speedup: 1.1×

• 43.9% 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: 94.2% 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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+225}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+225) (* x x) (fma x x (* y (fma x 2.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+225) {
		tmp = x * x;
	} else {
		tmp = fma(x, x, (y * fma(x, 2.0, y)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+225)
		tmp = Float64(x * x);
	else
		tmp = fma(x, x, Float64(y * fma(x, 2.0, y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+225], N[(x * x), $MachinePrecision], N[(x * x + N[(y * N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.000000000000001e225

   1. Initial program 86.7%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -6.000000000000001e225 < x

   1. Initial program 92.1%

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

      1. associate-+l+ N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-fma.f64 97.1

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

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-72}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -1.6e-72) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-72) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-1.6d-72)) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-72) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.6e-72:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-72)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e-72)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.6e-72], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e-72

   1. Initial program 91.5%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 75.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 75.1

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

   7. Applied egg-rr 75.1%

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

   if -1.6e-72 < x

   1. Initial program 91.9%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{y}^{2} + 0} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{y \cdot y} + 0 \]

      3. accelerator-lowering-fma.f64 60.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 0\right)} \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.9

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

   7. Applied egg-rr 60.9%

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

Alternative 3: 57.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x}^{2} + 0} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 60.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 60.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 60.0

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

7. Applied egg-rr 60.0%

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

Developer Target 1: 94.2% 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 2024195 
(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)))