Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.0% → 99.9%

Time: 3.5s

Alternatives: 4

Speedup: 0.5×

• 43.3% 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: 94.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x_m \cdot x_m \leq 5 \cdot 10^{+275}:\\ \;\;\;\;x_m \cdot x_m - y_m \cdot y_m\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \left(x_m + y_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y_m = (fabs.f64 y)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+275)
   (- (* 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+275) {
		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+275) 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+275) {
		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+275:
		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+275)
		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+275)
		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+275], 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) < 5.0000000000000003e275

   1. Initial program 100.0%

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

   if 5.0000000000000003e275 < (*.f64 x x)

   1. Initial program 78.3%

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

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 43.3%

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

      3. sqrt-prod 88.3%

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

      4. sqr-neg 88.3%

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

      5. sqrt-unprod 48.3%

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

      6. add-sqr-sqrt 85.0%

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

      7. sub-neg 85.0%

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

      8. pow1 85.0%

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

      9. pow1 85.0%

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

      10. pow-prod-up 85.0%

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

      11. add-sqr-sqrt 43.3%

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

      12. add-sqr-sqrt 13.3%

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

      13. difference-of-squares 13.3%

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

      14. metadata-eval 13.3%

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

      15. unpow-prod-down 13.3%

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

   4. Applied egg-rr 13.3%

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

      1. unpow2 13.3%

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

      2. unpow2 13.3%

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

      3. unswap-sqr 13.3%

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

      4. difference-of-squares 13.3%

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

      5. unpow1/2 13.3%

         \[\leadsto \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      6. unpow1/2 13.3%

         \[\leadsto \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      7. pow-sqr 13.3%

         \[\leadsto \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      8. metadata-eval 13.3%

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

      9. unpow1 13.3%

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

      10. unpow1/2 13.3%

          \[\leadsto \left(x - \color{blue}{{y}^{0.5}} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      11. unpow1/2 13.3%

          \[\leadsto \left(x - {y}^{0.5} \cdot \color{blue}{{y}^{0.5}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      12. pow-sqr 13.3%

          \[\leadsto \left(x - \color{blue}{{y}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      13. metadata-eval 13.3%

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

      14. unpow1 13.3%

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

      15. difference-of-squares 13.3%

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

      16. unpow1/2 13.3%

          \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \]

      17. unpow1/2 13.3%

          \[\leadsto \left(x - y\right) \cdot \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \]

      18. pow-sqr 36.7%

          \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \]

      19. metadata-eval 36.7%

          \[\leadsto \left(x - y\right) \cdot \left({x}^{\color{blue}{1}} - \sqrt{y} \cdot \sqrt{y}\right) \]

      20. unpow1 36.7%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in x around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l* 80.0%

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

      3. unpow2 80.0%

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

      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 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+275}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
y_m = abs(y)
function code(x_m, y_m)
	return fma(x_m, x_m, Float64(y_m * Float64(-y_m)))
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 * x$95$m + N[(y$95$m * (-y$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

   1. sqr-neg 94.9%

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

   2. cancel-sign-sub 94.9%

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

   3. fma-def 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right) \]
6. Add Preprocessing

Alternative 3: 61.8% 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 94.9%

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

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 43.6%

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

   3. sqrt-prod 76.2%

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

   4. sqr-neg 76.2%

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

   5. sqrt-unprod 33.2%

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

   6. add-sqr-sqrt 51.9%

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

   7. sub-neg 51.9%

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

   8. pow1 51.9%

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

   9. pow1 51.9%

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

   10. pow-prod-up 51.9%

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

   11. add-sqr-sqrt 26.5%

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

   12. add-sqr-sqrt 9.8%

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

   13. difference-of-squares 9.8%

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

   14. metadata-eval 9.8%

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

   15. unpow-prod-down 9.8%

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

4. Applied egg-rr 9.8%

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

   1. unpow2 9.8%

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

   2. unpow2 9.8%

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

   3. unswap-sqr 9.8%

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

   4. difference-of-squares 9.8%

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

   5. unpow1/2 9.8%

      \[\leadsto \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   6. unpow1/2 9.8%

      \[\leadsto \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   7. pow-sqr 9.8%

      \[\leadsto \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   8. metadata-eval 9.8%

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

   9. unpow1 9.8%

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

   10. unpow1/2 9.8%

       \[\leadsto \left(x - \color{blue}{{y}^{0.5}} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   11. unpow1/2 9.8%

       \[\leadsto \left(x - {y}^{0.5} \cdot \color{blue}{{y}^{0.5}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   12. pow-sqr 9.8%

       \[\leadsto \left(x - \color{blue}{{y}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   13. metadata-eval 9.8%

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

   14. unpow1 9.8%

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

   15. difference-of-squares 9.8%

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

   16. unpow1/2 9.8%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \]

   17. unpow1/2 9.8%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   18. pow-sqr 18.6%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   19. metadata-eval 18.6%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{\color{blue}{1}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   20. unpow1 18.6%

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

6. Simplified 51.9%

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

7. Taylor expanded in x around inf 53.2%

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

   1. *-commutative 53.2%

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

   2. associate-*l* 53.2%

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

   3. unpow2 53.2%

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

   4. distribute-lft-out 55.9%

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

9. Simplified 55.9%

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

10. Final simplification 55.9%

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

Alternative 4: 15.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (-2.0d0) * (x_m * y_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 43.6%

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

   3. sqrt-prod 76.2%

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

   4. sqr-neg 76.2%

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

   5. sqrt-unprod 33.2%

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

   6. add-sqr-sqrt 51.9%

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

   7. sub-neg 51.9%

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

   8. pow1 51.9%

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

   9. pow1 51.9%

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

   10. pow-prod-up 51.9%

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

   11. add-sqr-sqrt 26.5%

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

   12. add-sqr-sqrt 9.8%

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

   13. difference-of-squares 9.8%

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

   14. metadata-eval 9.8%

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

   15. unpow-prod-down 9.8%

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

4. Applied egg-rr 9.8%

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

   1. unpow2 9.8%

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

   2. unpow2 9.8%

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

   3. unswap-sqr 9.8%

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

   4. difference-of-squares 9.8%

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

   5. unpow1/2 9.8%

      \[\leadsto \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   6. unpow1/2 9.8%

      \[\leadsto \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   7. pow-sqr 9.8%

      \[\leadsto \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   8. metadata-eval 9.8%

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

   9. unpow1 9.8%

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

   10. unpow1/2 9.8%

       \[\leadsto \left(x - \color{blue}{{y}^{0.5}} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   11. unpow1/2 9.8%

       \[\leadsto \left(x - {y}^{0.5} \cdot \color{blue}{{y}^{0.5}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   12. pow-sqr 9.8%

       \[\leadsto \left(x - \color{blue}{{y}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   13. metadata-eval 9.8%

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

   14. unpow1 9.8%

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

   15. difference-of-squares 9.8%

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

   16. unpow1/2 9.8%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \]

   17. unpow1/2 9.8%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   18. pow-sqr 18.6%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   19. metadata-eval 18.6%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{\color{blue}{1}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   20. unpow1 18.6%

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

6. Simplified 51.9%

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

7. Taylor expanded in x around inf 53.2%

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

   1. *-commutative 53.2%

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

   2. associate-*l* 53.2%

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

   3. unpow2 53.2%

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

   4. distribute-lft-out 55.9%

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

9. Simplified 55.9%

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

10. Taylor expanded in x around 0 14.1%

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

11. Final simplification 14.1%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))