Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 5

Speedup: 1.0×

• 43.4% 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. Add Preprocessing

Alternative 2: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.5e+57) (* y (- x y)) (* x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.5e+57) {
		tmp = y * (x - y);
	} else {
		tmp = x * (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 <= 7.5d+57) then
        tmp = y * (x - y)
    else
        tmp = x * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.5e+57) {
		tmp = y * (x - y);
	} else {
		tmp = x * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.5e+57:
		tmp = y * (x - y)
	else:
		tmp = x * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.5e+57)
		tmp = Float64(y * Float64(x - y));
	else
		tmp = Float64(x * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.5e+57)
		tmp = y * (x - y);
	else
		tmp = x * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.5e+57], N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5000000000000006e57

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 7.5000000000000006e57 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 83.8%

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

      1. distribute-rgt-in 74.4%

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

      2. fma-define 74.4%

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

      3. *-commutative 74.4%

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

      4. add-sqr-sqrt 35.0%

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

      5. sqrt-unprod 78.8%

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

      6. sqr-neg 78.8%

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

      7. sqrt-unprod 43.8%

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

      8. add-sqr-sqrt 79.1%

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

      9. distribute-rgt-neg-out 79.1%

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

      10. fma-neg 74.4%

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

      11. pow2 74.4%

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

   7. Applied egg-rr 74.4%

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

      1. unpow2 74.4%

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

      2. distribute-lft-out-- 88.4%

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

   9. Simplified 88.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.2e+57) (* y (- y)) (* x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+57) {
		tmp = y * -y;
	} else {
		tmp = x * (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 <= 6.2d+57) then
        tmp = y * -y
    else
        tmp = x * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+57) {
		tmp = y * -y;
	} else {
		tmp = x * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.2e+57:
		tmp = y * -y
	else:
		tmp = x * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.2e+57)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.2e+57)
		tmp = y * -y;
	else
		tmp = x * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.2e+57], N[(y * (-y)), $MachinePrecision], N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000026e57

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 66.1%

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

      1. neg-mul-1 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in x around 0 64.3%

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

   if 6.20000000000000026e57 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 83.8%

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

      1. distribute-rgt-in 74.4%

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

      2. fma-define 74.4%

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

      3. *-commutative 74.4%

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

      4. add-sqr-sqrt 35.0%

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

      5. sqrt-unprod 78.8%

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

      6. sqr-neg 78.8%

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

      7. sqrt-unprod 43.8%

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

      8. add-sqr-sqrt 79.1%

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

      9. distribute-rgt-neg-out 79.1%

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

      10. fma-neg 74.4%

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

      11. pow2 74.4%

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

   7. Applied egg-rr 74.4%

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

      1. unpow2 74.4%

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

      2. distribute-lft-out-- 88.4%

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

   9. Simplified 88.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+57}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.22e+67) (* y (- y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.22e+67) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.22d+67) then
        tmp = y * -y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.22e+67) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.22e+67:
		tmp = y * -y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.22e+67)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.22e+67)
		tmp = y * -y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.22e+67], N[(y * (-y)), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.22000000000000004e67

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 66.2%

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

      1. neg-mul-1 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in x around 0 64.5%

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

   if 1.22000000000000004e67 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 85.1%

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

      1. distribute-rgt-in 75.6%

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

      2. fma-define 75.6%

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

      3. *-commutative 75.6%

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

      4. add-sqr-sqrt 35.6%

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

      5. sqrt-unprod 80.0%

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

      6. sqr-neg 80.0%

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

      7. sqrt-unprod 44.5%

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

      8. add-sqr-sqrt 78.7%

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

      9. distribute-rgt-neg-out 78.7%

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

      10. fma-neg 74.0%

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

      11. pow2 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. unpow2 74.0%

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

      2. distribute-lft-out-- 88.3%

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

   9. Simplified 88.3%

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

   10. Taylor expanded in x around inf 80.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 59.4%

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

   1. distribute-rgt-in 55.1%

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

   2. fma-define 55.9%

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

   3. *-commutative 55.9%

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

   4. add-sqr-sqrt 26.9%

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

   5. sqrt-unprod 63.3%

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

   6. sqr-neg 63.3%

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

   7. sqrt-unprod 30.2%

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

   8. add-sqr-sqrt 57.6%

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

   9. distribute-rgt-neg-out 57.6%

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

   10. fma-neg 56.0%

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

   11. pow2 56.0%

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

7. Applied egg-rr 56.0%

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

   1. unpow2 56.0%

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

   2. distribute-lft-out-- 61.1%

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

9. Simplified 61.1%

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

10. Taylor expanded in x around inf 56.7%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))