Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.7% → 99.5%

Time: 4.2s

Alternatives: 5

Speedup: 1.5×

• 42.4% 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: 93.7% 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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+220}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e+220) (* x x) (fma x x (* (fma 2.0 x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.8e+220) {
		tmp = x * x;
	} else {
		tmp = fma(x, x, (fma(2.0, x, y) * y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.8e+220)
		tmp = Float64(x * x);
	else
		tmp = fma(x, x, Float64(fma(2.0, x, y) * 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, -8.8e+220], N[(x * x), $MachinePrecision], N[(x * x + N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.79999999999999957e220

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -8.79999999999999957e220 < x

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 2: 88.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-84}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-84) (* x x) (* (fma 2.0 x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-84) {
		tmp = x * x;
	} else {
		tmp = fma(2.0, x, y) * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-84)
		tmp = Float64(x * x);
	else
		tmp = Float64(fma(2.0, x, y) * 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, -4.8e-84], N[(x * x), $MachinePrecision], N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000035e-84

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -4.80000000000000035e-84 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + 2 \cdot \frac{x}{y}\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(1 + 2 \cdot \frac{x}{y}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(1 + 2 \cdot \frac{x}{y}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(1 + 2 \cdot \frac{x}{y}\right)\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(1 + 2 \cdot \frac{x}{y}\right)\right) \cdot y} \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(2 \cdot \frac{x}{y} + 1\right)}\right) \cdot y \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \frac{x}{y}\right) \cdot y + 1 \cdot y\right)} \cdot y \]

      7. *-lft-identity N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

          \[\leadsto \left(\left(2 \cdot x\right) \cdot \color{blue}{1} + y\right) \cdot y \]

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 64.9

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

   5. Applied rewrites 64.9%

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

Alternative 3: 97.0% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma y y (* (fma y 2.0 x) x)))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(y, y, Float64(fma(y, 2.0, x) * x))
end

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 94.9

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 97.2

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

4. Applied rewrites 97.2%

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

Alternative 4: 88.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-84}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4.8e-84) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-84) {
		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 <= (-4.8d-84)) 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 <= -4.8e-84) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.8e-84:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-84)
		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 <= -4.8e-84)
		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, -4.8e-84], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000035e-84

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -4.80000000000000035e-84 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

Alternative 5: 56.9% 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 94.9%

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

3. Taylor expanded in y around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.6

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

5. Applied rewrites 57.6%

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

Developer Target 1: 93.7% 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 2024249 
(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)))