Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.9% → 99.8%

Time: 3.1s

Alternatives: 3

Speedup: 0.5×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+258}:\\ \;\;\;\;x\_m \cdot x\_m - y\_m \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m + y\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+258)
   (- (* x_m x_m) (* y_m y_m))
   (* x_m (+ x_m (* y_m -2.0)))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+258) {
		tmp = (x_m * x_m) - (y_m * y_m);
	} else {
		tmp = x_m * (x_m + (y_m * -2.0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x_m * x_m) <= 5d+258) then
        tmp = (x_m * x_m) - (y_m * y_m)
    else
        tmp = x_m * (x_m + (y_m * (-2.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	double tmp;
	if ((x_m * x_m) <= 5e+258) {
		tmp = (x_m * x_m) - (y_m * y_m);
	} else {
		tmp = x_m * (x_m + (y_m * -2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	tmp = 0
	if (x_m * x_m) <= 5e+258:
		tmp = (x_m * x_m) - (y_m * y_m)
	else:
		tmp = x_m * (x_m + (y_m * -2.0))
	return tmp

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 5e+258)
		tmp = Float64(Float64(x_m * x_m) - Float64(y_m * y_m));
	else
		tmp = Float64(x_m * Float64(x_m + Float64(y_m * -2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if ((x_m * x_m) <= 5e+258)
		tmp = (x_m * x_m) - (y_m * y_m);
	else
		tmp = x_m * (x_m + (y_m * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 5e+258], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m + N[(y$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5e258

   1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]
   2. Add Preprocessing

   if 5e258 < (*.f64 x x)

   1. Initial program 80.6%

      \[x \cdot x - y \cdot y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 73.6%

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

      2. pow2 73.6%

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

      3. difference-of-squares 81.9%

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

      4. sqrt-prod 38.8%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-prod 38.8%

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

      7. sqr-neg 38.8%

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

      8. sqrt-unprod 19.4%

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

      9. add-sqr-sqrt 38.8%

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

      10. sub-neg 38.8%

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

      11. add-sqr-sqrt 81.9%

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

      12. add-sqr-sqrt 38.8%

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

      13. add-sqr-sqrt 19.4%

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

      14. difference-of-squares 19.4%

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

      15. unpow-prod-down 19.4%

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

   4. Applied egg-rr 19.4%

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

      1. unpow2 19.4%

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

      2. unpow2 19.4%

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

      3. unswap-sqr 19.4%

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

      4. difference-of-squares 19.4%

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

      5. rem-square-sqrt 19.4%

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

      6. rem-square-sqrt 19.4%

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

      7. difference-of-squares 19.4%

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

      8. rem-square-sqrt 41.7%

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

      9. rem-square-sqrt 81.9%

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

   6. Simplified 81.9%

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

   7. Taylor expanded in x around inf 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l* 76.4%

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

      3. unpow2 76.4%

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

      4. distribute-lft-out 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ x\_m \cdot \left(x\_m + y\_m \cdot -2\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (* x_m (+ x_m (* y_m -2.0))))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return x_m * (x_m + (y_m * -2.0));
}

Fortran

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

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	return x_m * (x_m + (y_m * -2.0));
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return x_m * (x_m + (y_m * -2.0))

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(x_m * Float64(x_m + Float64(y_m * -2.0)))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = x_m * (x_m + (y_m * -2.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(x$95$m * N[(x$95$m + N[(y$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[x \cdot x - y \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 50.8%

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

   2. pow2 50.8%

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

   3. difference-of-squares 53.1%

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

   4. sqrt-prod 26.0%

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

   5. add-sqr-sqrt 14.0%

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

   6. sqrt-prod 26.7%

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

   7. sqr-neg 26.7%

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

   8. sqrt-unprod 13.1%

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

   9. add-sqr-sqrt 26.9%

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

   10. sub-neg 26.9%

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

   11. add-sqr-sqrt 53.2%

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

   12. add-sqr-sqrt 27.6%

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

   13. add-sqr-sqrt 15.1%

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

   14. difference-of-squares 15.1%

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

   15. unpow-prod-down 15.1%

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

4. Applied egg-rr 15.1%

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

   1. unpow2 15.1%

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

   2. unpow2 15.1%

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

   3. unswap-sqr 15.1%

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

   4. difference-of-squares 15.1%

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

   5. rem-square-sqrt 15.1%

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

   6. rem-square-sqrt 15.1%

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

   7. difference-of-squares 15.1%

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

   8. rem-square-sqrt 27.4%

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

   9. rem-square-sqrt 53.2%

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

6. Simplified 53.2%

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

7. Taylor expanded in x around inf 54.3%

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

   1. *-commutative 54.3%

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

   2. associate-*l* 54.3%

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

   3. unpow2 54.3%

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

   4. distribute-lft-out 58.6%

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

9. Simplified 58.6%

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

10. Final simplification 58.6%

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

Alternative 3: 15.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ y\_m \cdot \left(x\_m \cdot -2\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m) :precision binary64 (* y_m (* x_m -2.0)))

C

x_m = fabs(x);
y_m = fabs(y);
double code(double x_m, double y_m) {
	return y_m * (x_m * -2.0);
}

Fortran

x_m = abs(x)
y_m = abs(y)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = y_m * (x_m * (-2.0d0))
end function

Java

x_m = Math.abs(x);
y_m = Math.abs(y);
public static double code(double x_m, double y_m) {
	return y_m * (x_m * -2.0);
}

Python

x_m = math.fabs(x)
y_m = math.fabs(y)
def code(x_m, y_m):
	return y_m * (x_m * -2.0)

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return Float64(y_m * Float64(x_m * -2.0))
end

MATLAB

x_m = abs(x);
y_m = abs(y);
function tmp = code(x_m, y_m)
	tmp = y_m * (x_m * -2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
code[x$95$m_, y$95$m_] := N[(y$95$m * N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[x \cdot x - y \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 50.8%

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

   2. pow2 50.8%

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

   3. difference-of-squares 53.1%

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

   4. sqrt-prod 26.0%

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

   5. add-sqr-sqrt 14.0%

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

   6. sqrt-prod 26.7%

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

   7. sqr-neg 26.7%

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

   8. sqrt-unprod 13.1%

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

   9. add-sqr-sqrt 26.9%

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

   10. sub-neg 26.9%

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

   11. add-sqr-sqrt 53.2%

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

   12. add-sqr-sqrt 27.6%

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

   13. add-sqr-sqrt 15.1%

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

   14. difference-of-squares 15.1%

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

   15. unpow-prod-down 15.1%

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

4. Applied egg-rr 15.1%

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

   1. unpow2 15.1%

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

   2. unpow2 15.1%

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

   3. unswap-sqr 15.1%

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

   4. difference-of-squares 15.1%

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

   5. rem-square-sqrt 15.1%

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

   6. rem-square-sqrt 15.1%

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

   7. difference-of-squares 15.1%

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

   8. rem-square-sqrt 27.4%

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

   9. rem-square-sqrt 53.2%

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

6. Simplified 53.2%

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

7. Taylor expanded in x around inf 54.3%

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

   1. *-commutative 54.3%

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

   2. associate-*l* 54.3%

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

   3. unpow2 54.3%

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

   4. distribute-lft-out 58.6%

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

9. Simplified 58.6%

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

10. Taylor expanded in x around 0 17.1%

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

    1. associate-*r* 17.1%

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

    2. *-commutative 17.1%

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

12. Simplified 17.1%

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

13. Final simplification 17.1%

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

Reproduce

herbie shell --seed 2024039 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))