Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.6% → 98.3%

Time: 3.4s

Alternatives: 4

Speedup: 0.8×

• 43.9% 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: 93.6% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x_m, x_m, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x_m - y\right) \cdot \left(x_m - y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1e+206) (fma x_m x_m (* y (- y))) (* (- x_m y) (- x_m y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1e+206) {
		tmp = fma(x_m, x_m, (y * -y));
	} else {
		tmp = (x_m - y) * (x_m - y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1e+206)
		tmp = fma(x_m, x_m, Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x_m - y) * Float64(x_m - y));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1e+206], N[(x$95$m * x$95$m + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m - y), $MachinePrecision] * N[(x$95$m - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e206

   1. Initial program 93.7%

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

      1. sqr-neg 93.7%

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

      2. cancel-sign-sub 93.7%

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

      3. fma-def 97.0%

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

   3. Simplified 97.0%

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

   if 1e206 < x

   1. Initial program 78.9%

      \[x \cdot x - y \cdot y \]
   2. 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 68.4%

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

      3. sqrt-prod 94.7%

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

      4. sqr-neg 94.7%

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

      5. sqrt-unprod 26.3%

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

      6. add-sqr-sqrt 94.7%

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

      7. sub-neg 94.7%

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

      8. pow1 94.7%

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

      9. pow1 94.7%

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

      10. pow-prod-up 94.7%

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

      11. metadata-eval 94.7%

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

      12. add-sqr-sqrt 94.7%

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

      13. add-sqr-sqrt 68.4%

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

      14. difference-of-squares 68.4%

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

      15. unpow-prod-down 68.4%

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

   3. Applied egg-rr 68.4%

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

      1. unpow2 68.4%

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

      2. unpow2 68.4%

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

      3. unswap-sqr 68.4%

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

      4. difference-of-squares 68.4%

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

      5. unpow1/2 68.4%

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

      6. unpow1/2 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

      9. unpow1 68.4%

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

      10. unpow1/2 68.4%

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

      11. unpow1/2 68.4%

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

      12. pow-sqr 68.4%

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

      13. metadata-eval 68.4%

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

      14. unpow1 68.4%

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

      15. difference-of-squares 68.4%

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

      16. unpow1/2 68.4%

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

      17. unpow1/2 68.4%

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

      18. pow-sqr 68.4%

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

      19. metadata-eval 68.4%

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

      20. unpow1 68.4%

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

   5. Simplified 94.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x - y\right)\\ \end{array} \]

Alternative 2: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x_m - y\right) \cdot \left(x_m - y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1.9e-57) (* y (- y)) (* (- x_m y) (- x_m y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.9e-57) {
		tmp = y * -y;
	} else {
		tmp = (x_m - y) * (x_m - y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 1.9d-57) then
        tmp = y * -y
    else
        tmp = (x_m - y) * (x_m - y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.9e-57) {
		tmp = y * -y;
	} else {
		tmp = (x_m - y) * (x_m - y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 1.9e-57:
		tmp = y * -y
	else:
		tmp = (x_m - y) * (x_m - y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1.9e-57)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(Float64(x_m - y) * Float64(x_m - y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 1.9e-57)
		tmp = y * -y;
	else
		tmp = (x_m - y) * (x_m - y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1.9e-57], N[(y * (-y)), $MachinePrecision], N[(N[(x$95$m - y), $MachinePrecision] * N[(x$95$m - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999e-57

   1. Initial program 93.5%

      \[x \cdot x - y \cdot y \]

   2. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
   3. Step-by-step derivation

      1. mul-1-neg 62.0%

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

   4. Simplified 62.0%

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

      1. unpow2 62.0%

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

   6. Applied egg-rr 62.0%

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

   if 1.8999999999999999e-57 < x

   1. Initial program 90.1%

      \[x \cdot x - y \cdot y \]
   2. 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 52.1%

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

      3. sqrt-prod 88.5%

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

      4. sqr-neg 88.5%

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

      5. sqrt-unprod 36.4%

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

      6. add-sqr-sqrt 74.8%

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

      7. sub-neg 74.8%

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

      8. pow1 74.8%

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

      9. pow1 74.8%

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

      10. pow-prod-up 74.8%

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

      11. metadata-eval 74.8%

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

      12. add-sqr-sqrt 74.4%

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

      13. add-sqr-sqrt 38.2%

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

      14. difference-of-squares 38.2%

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

      15. unpow-prod-down 38.2%

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

   3. Applied egg-rr 38.2%

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

      1. unpow2 38.2%

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

      2. unpow2 38.2%

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

      3. unswap-sqr 38.2%

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

      4. difference-of-squares 38.2%

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

      5. unpow1/2 38.2%

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

      6. unpow1/2 38.2%

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

      7. pow-sqr 38.3%

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

      8. metadata-eval 38.3%

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

      9. unpow1 38.3%

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

      10. unpow1/2 38.3%

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

      11. unpow1/2 38.3%

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

      12. pow-sqr 38.3%

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

      13. metadata-eval 38.3%

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

      14. unpow1 38.3%

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

      15. difference-of-squares 38.3%

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

      16. unpow1/2 38.3%

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

      17. unpow1/2 38.3%

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

      18. pow-sqr 38.4%

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

      19. metadata-eval 38.4%

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

      20. unpow1 38.4%

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

   5. Simplified 74.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x - y\right)\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;x_m \cdot x_m - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x_m - y\right) \cdot \left(x_m - y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1.18e+147) (- (* x_m x_m) (* y y)) (* (- x_m y) (- x_m y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.18e+147) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = (x_m - y) * (x_m - y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 1.18d+147) then
        tmp = (x_m * x_m) - (y * y)
    else
        tmp = (x_m - y) * (x_m - y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.18e+147) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = (x_m - y) * (x_m - y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 1.18e+147:
		tmp = (x_m * x_m) - (y * y)
	else:
		tmp = (x_m - y) * (x_m - y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1.18e+147)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * y));
	else
		tmp = Float64(Float64(x_m - y) * Float64(x_m - y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 1.18e+147)
		tmp = (x_m * x_m) - (y * y);
	else
		tmp = (x_m - y) * (x_m - y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1.18e+147], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m - y), $MachinePrecision] * N[(x$95$m - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18000000000000006e147

   1. Initial program 94.6%

      \[x \cdot x - y \cdot y \]

   if 1.18000000000000006e147 < x

   1. Initial program 78.8%

      \[x \cdot x - y \cdot y \]
   2. 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 51.5%

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

      3. sqrt-prod 90.9%

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

      4. sqr-neg 90.9%

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

      5. sqrt-unprod 39.4%

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

      6. add-sqr-sqrt 87.9%

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

      7. sub-neg 87.9%

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

      8. pow1 87.9%

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

      9. pow1 87.9%

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

      10. pow-prod-up 87.9%

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

      11. metadata-eval 87.9%

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

      12. add-sqr-sqrt 87.9%

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

      13. add-sqr-sqrt 48.5%

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

      14. difference-of-squares 48.5%

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

      15. unpow-prod-down 48.5%

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

   3. Applied egg-rr 48.5%

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

      1. unpow2 48.5%

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

      2. unpow2 48.5%

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

      3. unswap-sqr 48.5%

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

      4. difference-of-squares 48.5%

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

      5. unpow1/2 48.5%

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

      6. unpow1/2 48.5%

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

      7. pow-sqr 48.5%

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

      8. metadata-eval 48.5%

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

      9. unpow1 48.5%

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

      10. unpow1/2 48.5%

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

      11. unpow1/2 48.5%

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

      12. pow-sqr 48.5%

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

      13. metadata-eval 48.5%

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

      14. unpow1 48.5%

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

      15. difference-of-squares 48.5%

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

      16. unpow1/2 48.5%

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

      17. unpow1/2 48.5%

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

      18. pow-sqr 48.5%

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

      19. metadata-eval 48.5%

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

      20. unpow1 48.5%

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

   5. Simplified 87.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x - y\right)\\ \end{array} \]

Alternative 4: 52.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

   \[x \cdot x - y \cdot y \]

2. Taylor expanded in x around 0 51.8%

   \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
3. Step-by-step derivation

   1. mul-1-neg 51.8%

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

4. Simplified 51.8%

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

   1. unpow2 51.8%

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

6. Applied egg-rr 51.8%

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

7. Final simplification 51.8%

   \[\leadsto y \cdot \left(-y\right) \]

Reproduce

herbie shell --seed 2023315 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))