Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.2% → 99.7%

Time: 12.4s

Alternatives: 6

Speedup: 0.5×

• 42.5% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 4e+300)
   (fma x x (* y (+ (* x 2.0) y)))
   (* y (+ y (* x (+ 2.0 (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 4e+300) {
		tmp = fma(x, x, (y * ((x * 2.0) + y)));
	} else {
		tmp = y * (y + (x * (2.0 + (x / y))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 4e+300)
		tmp = fma(x, x, Float64(y * Float64(Float64(x * 2.0) + y)));
	else
		tmp = Float64(y * Float64(y + Float64(x * Float64(2.0 + Float64(x / 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[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 4e+300], N[(x * x + N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * N[(2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y)) < 4.0000000000000002e300

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*l* 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 4.0000000000000002e300 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   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 y around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. associate-/l* 91.6%

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

      4. distribute-lft-out 91.6%

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

   5. Simplified 91.6%

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

      1. associate-*r* 92.4%

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

      2. +-commutative 92.4%

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

      3. distribute-rgt-in 86.7%

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

      4. *-commutative 86.7%

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

      5. *-commutative 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. distribute-rgt-out 92.4%

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

      2. *-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. *-commutative 92.4%

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

   9. Applied egg-rr 92.4%

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

       1. expm1-log1p-u 92.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)\right)} \]

       2. expm1-undefine 92.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)} - 1} \]

       3. fma-define 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{x}{y}, y \cdot x, y \cdot y\right)}\right)} - 1 \]

       4. *-commutative 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, \color{blue}{x \cdot y}, y \cdot y\right)\right)} - 1 \]

       5. pow2 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, \color{blue}{{y}^{2}}\right)\right)} - 1 \]

   11. Applied egg-rr 92.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. log1p-undefine 92.2%

          \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)}} - 1 \]

       2. rem-exp-log 92.4%

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

       3. +-commutative 92.4%

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

       4. associate--l+ 92.4%

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

       5. metadata-eval 92.4%

          \[\leadsto \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right) + \color{blue}{0} \]

       6. +-rgt-identity 92.4%

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

       7. fma-undefine 92.4%

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

       8. +-commutative 92.4%

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

       9. unpow2 92.4%

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

       10. associate-*r* 91.6%

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

       11. *-commutative 91.6%

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

       12. distribute-rgt-out 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 4 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 4 \cdot 10^{+300}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 4e+300)
   (+ (* y y) (* x (+ x (* 2.0 y))))
   (* y (+ y (* x (+ 2.0 (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 4e+300) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * (y + (x * (2.0 + (x / 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 * x) + ((x * 2.0d0) * y)) + (y * y)) <= 4d+300) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = y * (y + (x * (2.0d0 + (x / y))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 4e+300) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * (y + (x * (2.0 + (x / y))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (((x * x) + ((x * 2.0) * y)) + (y * y)) <= 4e+300:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = y * (y + (x * (2.0 + (x / y))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 4e+300)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(y * Float64(y + Float64(x * Float64(2.0 + Float64(x / y)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 4e+300)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = y * (y + (x * (2.0 + (x / 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[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 4e+300], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * N[(2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y)) < 4.0000000000000002e300

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. distribute-lft-out 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 4.0000000000000002e300 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   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 y around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. unpow2 93.5%

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

      3. associate-/l* 91.6%

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

      4. distribute-lft-out 91.6%

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

   5. Simplified 91.6%

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

      1. associate-*r* 92.4%

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

      2. +-commutative 92.4%

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

      3. distribute-rgt-in 86.7%

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

      4. *-commutative 86.7%

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

      5. *-commutative 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. distribute-rgt-out 92.4%

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

      2. *-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. *-commutative 92.4%

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

   9. Applied egg-rr 92.4%

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

       1. expm1-log1p-u 92.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)\right)} \]

       2. expm1-undefine 92.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)} - 1} \]

       3. fma-define 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{x}{y}, y \cdot x, y \cdot y\right)}\right)} - 1 \]

       4. *-commutative 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, \color{blue}{x \cdot y}, y \cdot y\right)\right)} - 1 \]

       5. pow2 92.2%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, \color{blue}{{y}^{2}}\right)\right)} - 1 \]

   11. Applied egg-rr 92.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. log1p-undefine 92.2%

          \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)}} - 1 \]

       2. rem-exp-log 92.4%

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

       3. +-commutative 92.4%

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

       4. associate--l+ 92.4%

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

       5. metadata-eval 92.4%

          \[\leadsto \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right) + \color{blue}{0} \]

       6. +-rgt-identity 92.4%

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

       7. fma-undefine 92.4%

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

       8. +-commutative 92.4%

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

       9. unpow2 92.4%

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

       10. associate-*r* 91.6%

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

       11. *-commutative 91.6%

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

       12. distribute-rgt-out 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 4 \cdot 10^{+300}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return y * (y + (x * (2.0 + (x / 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 + (x * (2.0d0 + (x / y))))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y * (y + (x * (2.0 + (x / y))));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * (y + (x * (2.0 + (x / y))))

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * (y + (x * (2.0 + (x / y))));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * N[(y + N[(x * N[(2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 y around inf 88.9%

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

   1. *-commutative 88.9%

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

   2. unpow2 88.9%

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

   3. associate-/l* 88.0%

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

   4. distribute-lft-out 88.0%

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

5. Simplified 88.0%

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

   1. associate-*r* 88.3%

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

   2. +-commutative 88.3%

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

   3. distribute-rgt-in 86.0%

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

   4. *-commutative 86.0%

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

   5. *-commutative 86.0%

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

7. Applied egg-rr 86.0%

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

   1. distribute-rgt-out 88.3%

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

   2. *-commutative 88.3%

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

   3. +-commutative 88.3%

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

   4. *-commutative 88.3%

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

9. Applied egg-rr 88.3%

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

    1. expm1-log1p-u 85.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)\right)} \]

    2. expm1-undefine 75.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 + \frac{x}{y}\right) \cdot \left(y \cdot x\right) + y \cdot y\right)} - 1} \]

    3. fma-define 75.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{x}{y}, y \cdot x, y \cdot y\right)}\right)} - 1 \]

    4. *-commutative 75.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, \color{blue}{x \cdot y}, y \cdot y\right)\right)} - 1 \]

    5. pow2 75.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, \color{blue}{{y}^{2}}\right)\right)} - 1 \]

11. Applied egg-rr 75.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)} - 1} \]
12. Step-by-step derivation

    1. log1p-undefine 75.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right)\right)}} - 1 \]

    2. rem-exp-log 78.5%

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

    3. +-commutative 78.5%

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

    4. associate--l+ 88.3%

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

    5. metadata-eval 88.3%

       \[\leadsto \mathsf{fma}\left(2 + \frac{x}{y}, x \cdot y, {y}^{2}\right) + \color{blue}{0} \]

    6. +-rgt-identity 88.3%

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

    7. fma-undefine 88.3%

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

    8. +-commutative 88.3%

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

    9. unpow2 88.3%

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

    10. associate-*r* 88.0%

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

    11. *-commutative 88.0%

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

    12. distribute-rgt-out 91.1%

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

13. Simplified 91.1%

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

14. Final simplification 91.1%

    \[\leadsto y \cdot \left(y + x \cdot \left(2 + \frac{x}{y}\right)\right) \]
15. Add Preprocessing

Alternative 4: 57.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return y * ((x * 2.0) + 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 * ((x * 2.0d0) + y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y * ((x * 2.0) + y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y * ((x * 2.0) + y)

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y * ((x * 2.0) + y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $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 53.0%

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

   1. *-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. associate-*r* 53.0%

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

   4. *-commutative 53.0%

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

5. Simplified 53.0%

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

6. Taylor expanded in y around 0 56.1%

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

7. Final simplification 56.1%

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

Alternative 5: 15.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 2.0 * (x * 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 = 2.0d0 * (x * y)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return 2.0 * (x * y)

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 2.0 * (x * y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(2.0 * N[(x * y), $MachinePrecision]), $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 53.0%

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

   1. *-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. associate-*r* 53.0%

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

   4. *-commutative 53.0%

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

5. Simplified 53.0%

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

6. Taylor expanded in y around 0 56.1%

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

7. Taylor expanded in y around 0 11.8%

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

8. Final simplification 11.8%

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

Alternative 6: 15.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (x * 2.0) * 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 = (x * 2.0d0) * y
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x * 2.0) * y

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x * 2.0) * y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x * 2.0), $MachinePrecision] * y), $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 53.0%

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

   1. *-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. associate-*r* 53.0%

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

   4. *-commutative 53.0%

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

5. Simplified 53.0%

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

6. Taylor expanded in y around 0 56.1%

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

7. Taylor expanded in y around 0 11.8%

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

   1. associate-*r* 11.8%

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

9. Simplified 11.8%

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

10. Final simplification 11.8%

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

Developer target: 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 2024076 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))