Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.5% → 100.0%

Time: 3.8s

Alternatives: 6

Speedup: 1.5×

• 43.4% 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: 93.5% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. 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. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(y, y, \mathsf{fma}\left(y, 2, x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e+123) (fma y y (* (fma y 2.0 x) x)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.5e+123) {
		tmp = fma(y, y, (fma(y, 2.0, x) * x));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.5e+123)
		tmp = fma(y, y, Float64(fma(y, 2.0, x) * x));
	else
		tmp = Float64(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[y, 2.5e+123], N[(y * y + N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999987e123

   1. Initial program 95.0%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 95.0

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

      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 97.7

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

   4. Applied rewrites 97.7%

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

   if 2.49999999999999987e123 < y

   1. Initial program 89.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 3: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 2, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, \left(y \cdot 2\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-8) (fma (* x 2.0) y (* x x)) (fma y y (* (* y 2.0) x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-8) {
		tmp = fma((x * 2.0), y, (x * x));
	} else {
		tmp = fma(y, y, ((y * 2.0) * x));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-8)
		tmp = fma(Float64(x * 2.0), y, Float64(x * x));
	else
		tmp = fma(y, y, Float64(Float64(y * 2.0) * x));
	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-8], N[(N[(x * 2.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(y * y + N[(N[(y * 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999997e-8

   1. Initial program 88.7%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -4.79999999999999997e-8 < 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. +-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 95.9

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

      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 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 2, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, \left(y \cdot 2\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, \left(y \cdot 2\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-8) (* (fma y 2.0 x) x) (fma y y (* (* y 2.0) x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-8) {
		tmp = fma(y, 2.0, x) * x;
	} else {
		tmp = fma(y, y, ((y * 2.0) * x));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-8)
		tmp = Float64(fma(y, 2.0, x) * x);
	else
		tmp = fma(y, y, Float64(Float64(y * 2.0) * x));
	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-8], N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision], N[(y * y + N[(N[(y * 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999997e-8

   1. Initial program 88.7%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 17.7

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. *-inverses N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if -4.79999999999999997e-8 < 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. +-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 95.9

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

      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 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

Alternative 5: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 2, x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-8) (* (fma y 2.0 x) x) (* (fma 2.0 x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-8) {
		tmp = fma(y, 2.0, 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-8)
		tmp = Float64(fma(y, 2.0, 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-8], N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999997e-8

   1. Initial program 88.7%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 17.7

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. *-inverses N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if -4.79999999999999997e-8 < 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. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{x \cdot {y}^{2}}{y}} + {y}^{2} \]

      6. associate-*l/ N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 71.1%

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

Alternative 6: 57.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

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 = y * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
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 57.4

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

5. Applied rewrites 57.4%

   \[\leadsto \color{blue}{y \cdot y} \]
6. 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 2024296 
(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)))