Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.5% → 99.9%

Time: 4.2s

Alternatives: 6

Speedup: 1.1×

• 44.1% 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: 93.5% 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: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, y, \mathsf{fma}\left(y, 2, x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e+148) (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 <= 2.25e+148) {
		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 <= 2.25e+148)
		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, 2.25e+148], 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 < 2.24999999999999997e148

   1. Initial program 95.4%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 95.4

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   if 2.24999999999999997e148 < y

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 2, x\right) \cdot x\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, 2, y \cdot y\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 2.0 x) x)))
   (if (<= y 2.8e-114)
     t_0
     (if (<= y 3.6e-78)
       (fma (* y x) 2.0 (* y y))
       (if (<= y 5.6e-26) t_0 (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = fma(y, 2.0, x) * x;
	double tmp;
	if (y <= 2.8e-114) {
		tmp = t_0;
	} else if (y <= 3.6e-78) {
		tmp = fma((y * x), 2.0, (y * y));
	} else if (y <= 5.6e-26) {
		tmp = t_0;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(fma(y, 2.0, x) * x)
	tmp = 0.0
	if (y <= 2.8e-114)
		tmp = t_0;
	elseif (y <= 3.6e-78)
		tmp = fma(Float64(y * x), 2.0, Float64(y * y));
	elseif (y <= 5.6e-26)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, 2.8e-114], t$95$0, If[LessEqual[y, 3.6e-78], N[(N[(y * x), $MachinePrecision] * 2.0 + N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-26], t$95$0, N[(y * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.8000000000000001e-114 or 3.6000000000000002e-78 < y < 5.6000000000000002e-26

   1. Initial program 95.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 50.1

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

   5. Applied rewrites 50.1%

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

   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. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. *-inverses N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative 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 66.2

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

   8. Applied rewrites 66.2%

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

   if 2.8000000000000001e-114 < y < 3.6000000000000002e-78

   1. Initial program 100.0%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 62.0%

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

   7. Applied rewrites 62.0%

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

   if 5.6000000000000002e-26 < y

   1. Initial program 91.3%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 3: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 2, x\right) \cdot x\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y\right) \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma y 2.0 x) x)))
   (if (<= y 2.8e-114)
     t_0
     (if (<= y 3.6e-78) (* (fma 2.0 x y) y) (if (<= y 5.6e-26) t_0 (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = fma(y, 2.0, x) * x;
	double tmp;
	if (y <= 2.8e-114) {
		tmp = t_0;
	} else if (y <= 3.6e-78) {
		tmp = fma(2.0, x, y) * y;
	} else if (y <= 5.6e-26) {
		tmp = t_0;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(fma(y, 2.0, x) * x)
	tmp = 0.0
	if (y <= 2.8e-114)
		tmp = t_0;
	elseif (y <= 3.6e-78)
		tmp = Float64(fma(2.0, x, y) * y);
	elseif (y <= 5.6e-26)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y * 2.0 + x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, 2.8e-114], t$95$0, If[LessEqual[y, 3.6e-78], N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 5.6e-26], t$95$0, N[(y * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.8000000000000001e-114 or 3.6000000000000002e-78 < y < 5.6000000000000002e-26

   1. Initial program 95.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 50.1

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

   5. Applied rewrites 50.1%

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

   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. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. *-inverses N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative 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 66.2

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

   8. Applied rewrites 66.2%

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

   if 2.8000000000000001e-114 < y < 3.6000000000000002e-78

   1. Initial program 100.0%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 62.0%

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

   if 5.6000000000000002e-26 < y

   1. Initial program 91.3%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 4: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-114} \lor \neg \left(y \leq 3.6 \cdot 10^{-78} \lor \neg \left(y \leq 4.2 \cdot 10^{-23}\right)\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= y 2.8e-114) (not (or (<= y 3.6e-78) (not (<= y 4.2e-23)))))
   (* x x)
   (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((y <= 2.8e-114) || !((y <= 3.6e-78) || !(y <= 4.2e-23))) {
		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 ((y <= 2.8d-114) .or. (.not. (y <= 3.6d-78) .or. (.not. (y <= 4.2d-23)))) 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 ((y <= 2.8e-114) || !((y <= 3.6e-78) || !(y <= 4.2e-23))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (y <= 2.8e-114) or not ((y <= 3.6e-78) or not (y <= 4.2e-23)):
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((y <= 2.8e-114) || !((y <= 3.6e-78) || !(y <= 4.2e-23)))
		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 ((y <= 2.8e-114) || ~(((y <= 3.6e-78) || ~((y <= 4.2e-23)))))
		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[Or[LessEqual[y, 2.8e-114], N[Not[Or[LessEqual[y, 3.6e-78], N[Not[LessEqual[y, 4.2e-23]], $MachinePrecision]]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8000000000000001e-114 or 3.6000000000000002e-78 < y < 4.2000000000000002e-23

   1. Initial program 95.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   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 65.8

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

   8. Applied rewrites 65.8%

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

   if 2.8000000000000001e-114 < y < 3.6000000000000002e-78 or 4.2000000000000002e-23 < y

   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 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-114} \lor \neg \left(y \leq 3.6 \cdot 10^{-78} \lor \neg \left(y \leq 4.2 \cdot 10^{-23}\right)\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y\right) \cdot y\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-114)
   (* x x)
   (if (<= y 3.6e-78)
     (* (fma 2.0 x y) y)
     (if (<= y 4.2e-23) (* x x) (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-114) {
		tmp = x * x;
	} else if (y <= 3.6e-78) {
		tmp = fma(2.0, x, y) * y;
	} else if (y <= 4.2e-23) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-114)
		tmp = Float64(x * x);
	elseif (y <= 3.6e-78)
		tmp = Float64(fma(2.0, x, y) * y);
	elseif (y <= 4.2e-23)
		tmp = Float64(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, 2.8e-114], N[(x * x), $MachinePrecision], If[LessEqual[y, 3.6e-78], N[(N[(2.0 * x + y), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4.2e-23], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.8000000000000001e-114 or 3.6000000000000002e-78 < y < 4.2000000000000002e-23

   1. Initial program 95.6%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   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 65.8

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

   8. Applied rewrites 65.8%

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

   if 2.8000000000000001e-114 < y < 3.6000000000000002e-78

   1. Initial program 100.0%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 62.0%

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

   if 4.2000000000000002e-23 < y

   1. Initial program 91.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

Alternative 6: 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 94.5%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 60.3

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

5. Applied rewrites 60.3%

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

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 54.8

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

8. Applied rewrites 54.8%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024324 
(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)))