Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 7

Speedup: 1.0×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x + y)
end function

Java

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

Python

def code(x, y):
	return (x + y) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x + y)
end function

Java

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

Python

def code(x, y):
	return (x + y) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) * (x + y)
end function

Java

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

Python

def code(x, y):
	return (x + y) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e-150) (* x (+ x (* y 2.0))) (* y (- y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-150) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d-150) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * (y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-150) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e-150:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * (y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-150)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * Float64(y - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-150)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * (y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e-150], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.60000000000000006e-150

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*l* 66.6%

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

      3. unpow2 66.6%

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

      4. distribute-lft-out 67.8%

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

   5. Simplified 67.8%

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

   if 4.60000000000000006e-150 < y

   1. Initial program 100.0%

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

      1. flip-+ 93.4%

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

      2. associate-*r/ 78.2%

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

      3. pow2 78.2%

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

      4. pow2 78.2%

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

   4. Applied egg-rr 78.2%

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

      1. associate-/l* 93.3%

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

      2. +-commutative 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in x around 0 74.5%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   9. Applied egg-rr 73.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 15.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-310) (* x (- y)) (* 2.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-310) {
		tmp = x * -y;
	} else {
		tmp = 2.0 * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = x * -y
    else
        tmp = 2.0d0 * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-310) {
		tmp = x * -y;
	} else {
		tmp = 2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-310:
		tmp = x * -y
	else:
		tmp = 2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = x * -y;
	else
		tmp = 2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-310], N[(x * (-y)), $MachinePrecision], N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 100.0%

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

      1. flip-+ 93.3%

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

      2. associate-*r/ 77.7%

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

      3. pow2 77.7%

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

      4. pow2 77.7%

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

   4. Applied egg-rr 77.7%

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

      1. associate-/l* 93.1%

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

      2. +-commutative 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in x around 0 52.7%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   9. Applied egg-rr 52.2%

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

   10. Taylor expanded in y around 0 11.3%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 11.3%

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

       2. distribute-lft-neg-in 11.3%

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

       3. *-commutative 11.3%

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

   12. Simplified 11.3%

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

   if -9.999999999999969e-311 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-*l* 52.4%

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

      3. unpow2 52.4%

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

      4. distribute-lft-out 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around 0 15.9%

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

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e+220) (* x (- y)) (* y (+ x y))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.8d+220)) then
        tmp = x * -y
    else
        tmp = y * (x + y)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -8.8e+220:
		tmp = x * -y
	else:
		tmp = y * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.8e+220)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(y * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.8e+220)
		tmp = x * -y;
	else
		tmp = y * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.8e+220], N[(x * (-y)), $MachinePrecision], N[(y * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.79999999999999957e220

   1. Initial program 100.0%

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

      1. flip-+ 77.8%

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

      2. associate-*r/ 77.8%

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

      3. pow2 77.8%

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

      4. pow2 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. associate-/l* 77.8%

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

      2. +-commutative 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in x around 0 18.3%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   9. Applied egg-rr 18.8%

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

   10. Taylor expanded in y around 0 18.8%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 18.8%

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

       2. distribute-lft-neg-in 18.8%

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

       3. *-commutative 18.8%

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

   12. Simplified 18.8%

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

   if -8.79999999999999957e220 < x

   1. Initial program 100.0%

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

      1. flip-+ 96.2%

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

      2. associate-*r/ 80.2%

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

      3. pow2 80.2%

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

      4. pow2 80.2%

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

   4. Applied egg-rr 80.2%

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

      1. associate-/l* 96.1%

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

      2. +-commutative 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in x around 0 60.4%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 60.5%

         \[\leadsto \color{blue}{\frac{y + x}{1} \cdot y} \]

      2. /-rgt-identity 60.5%

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

      3. +-commutative 60.5%

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

   9. Applied egg-rr 60.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+220}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e-303) (* 2.0 (* x y)) (* y (- y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e-303) {
		tmp = 2.0 * (x * y);
	} else {
		tmp = y * (y - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8d-303)) then
        tmp = 2.0d0 * (x * y)
    else
        tmp = y * (y - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e-303) {
		tmp = 2.0 * (x * y);
	} else {
		tmp = y * (y - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e-303:
		tmp = 2.0 * (x * y)
	else:
		tmp = y * (y - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e-303)
		tmp = Float64(2.0 * Float64(x * y));
	else
		tmp = Float64(y * Float64(y - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e-303)
		tmp = 2.0 * (x * y);
	else
		tmp = y * (y - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e-303], N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999944e-303

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-*l* 60.7%

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

      3. unpow2 60.7%

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

      4. distribute-lft-out 62.1%

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x around 0 15.4%

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

   if -7.99999999999999944e-303 < y

   1. Initial program 100.0%

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

      1. flip-+ 94.8%

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

      2. associate-*r/ 80.0%

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

      3. pow2 80.0%

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

      4. pow2 80.0%

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

   4. Applied egg-rr 80.0%

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

      1. associate-/l* 94.7%

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

      2. +-commutative 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in x around 0 63.5%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   9. Applied egg-rr 62.7%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 15.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1e-310) (* x (- y)) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = x * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-310) {
		tmp = x * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-310:
		tmp = x * -y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = x * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-310], N[(x * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 100.0%

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

      1. flip-+ 93.3%

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

      2. associate-*r/ 77.7%

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

      3. pow2 77.7%

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

      4. pow2 77.7%

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

   4. Applied egg-rr 77.7%

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

      1. associate-/l* 93.1%

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

      2. +-commutative 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in x around 0 52.7%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   9. Applied egg-rr 52.2%

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

   10. Taylor expanded in y around 0 11.3%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 11.3%

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

       2. distribute-lft-neg-in 11.3%

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

       3. *-commutative 11.3%

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

   12. Simplified 11.3%

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

   if -9.999999999999969e-311 < x

   1. Initial program 100.0%

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

      1. flip-+ 96.3%

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

      2. associate-*r/ 82.1%

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

      3. pow2 82.1%

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

      4. pow2 82.1%

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

   4. Applied egg-rr 82.1%

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

      1. associate-/l* 96.2%

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

      2. +-commutative 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

   8. Taylor expanded in y around 0 15.9%

      \[\leadsto \color{blue}{x \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative 15.9%

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

   10. Simplified 15.9%

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

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 15.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * y
end function

Java

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

Python

def code(x, y):
	return x * y

Julia

function code(x, y)
	return Float64(x * y)
end

MATLAB

function tmp = code(x, y)
	tmp = x * y;
end

Wolfram

code[x_, y_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. flip-+ 94.9%

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

   2. associate-*r/ 80.1%

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

   3. pow2 80.1%

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

   4. pow2 80.1%

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

4. Applied egg-rr 80.1%

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

   1. associate-/l* 94.8%

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

   2. +-commutative 94.8%

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

6. Simplified 94.8%

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

7. Taylor expanded in x around 0 57.4%

   \[\leadsto \frac{y + x}{\color{blue}{\frac{1}{y}}} \]

8. Taylor expanded in y around 0 14.3%

   \[\leadsto \color{blue}{x \cdot y} \]
9. Step-by-step derivation

   1. *-commutative 14.3%

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

10. Simplified 14.3%

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

11. Final simplification 14.3%

    \[\leadsto x \cdot y \]
12. Add Preprocessing

Developer target: 93.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (+ (* y y) (* 2.0 (* y x)))))

C

double code(double x, double y) {
	return (x * x) + ((y * y) + (2.0 * (y * x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) + (2.0d0 * (y * x)))
end function

Java

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

Python

def code(x, y):
	return (x * x) + ((y * y) + (2.0 * (y * x)))

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(2.0 * Float64(y * x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + (2.0 * (y * x)));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024014 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

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