Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.5% → 100.0%

Time: 6.5s

Alternatives: 4

Speedup: 1.0×

• 44.8% 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: 93.5% 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: 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 94.1%

   \[x \cdot x - y \cdot y \]
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. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

   1. add-sqr-sqrt 28.2%

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

   2. sqrt-prod 75.5%

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

   3. add-sqr-sqrt 28.2%

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

   4. add-sqr-sqrt 28.2%

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

   5. sqr-neg 28.2%

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

   6. swap-sqr 28.2%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 51.4%

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

   9. distribute-rgt-neg-out 51.4%

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

   10. add-sqr-sqrt 100.0%

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

   11. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-185}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-61}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-185)
   (* x x)
   (if (<= (* y y) 2e-133)
     (* y (- y))
     (if (<= (* y y) 2e-61) (* x x) (* y (- x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-185) {
		tmp = x * x;
	} else if ((y * y) <= 2e-133) {
		tmp = y * -y;
	} else if ((y * y) <= 2e-61) {
		tmp = x * x;
	} else {
		tmp = y * (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 2d-185) then
        tmp = x * x
    else if ((y * y) <= 2d-133) then
        tmp = y * -y
    else if ((y * y) <= 2d-61) then
        tmp = x * x
    else
        tmp = y * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-185) {
		tmp = x * x;
	} else if ((y * y) <= 2e-133) {
		tmp = y * -y;
	} else if ((y * y) <= 2e-61) {
		tmp = x * x;
	} else {
		tmp = y * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-185:
		tmp = x * x
	elif (y * y) <= 2e-133:
		tmp = y * -y
	elif (y * y) <= 2e-61:
		tmp = x * x
	else:
		tmp = y * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-185)
		tmp = Float64(x * x);
	elseif (Float64(y * y) <= 2e-133)
		tmp = Float64(y * Float64(-y));
	elseif (Float64(y * y) <= 2e-61)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e-185)
		tmp = x * x;
	elseif ((y * y) <= 2e-133)
		tmp = y * -y;
	elseif ((y * y) <= 2e-61)
		tmp = x * x;
	else
		tmp = y * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-185], N[(x * x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e-133], N[(y * (-y)), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e-61], N[(x * x), $MachinePrecision], N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2e-185 or 2.0000000000000001e-133 < (*.f64 y y) < 2.0000000000000001e-61

   1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 45.1%

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

      4. sqrt-unprod 93.7%

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

      5. sqr-neg 93.7%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 91.2%

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in x around inf 91.4%

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

   6. Taylor expanded in x around inf 91.9%

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

   if 2e-185 < (*.f64 y y) < 2.0000000000000001e-133

   1. Initial program 99.9%

      \[x \cdot x - y \cdot y \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 43.4%

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

      4. sqrt-unprod 57.7%

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

      5. sqr-neg 57.7%

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

      6. sqrt-prod 13.9%

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

      7. add-sqr-sqrt 27.4%

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

   4. Applied egg-rr 27.4%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-prod 57.7%

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

      3. add-sqr-sqrt 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqr-neg 13.9%

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

      6. swap-sqr 13.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 55.8%

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

      9. distribute-rgt-neg-out 55.8%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 72.5%

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

   8. Taylor expanded in x around 0 74.2%

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

      1. neg-mul-1 74.2%

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

   10. Simplified 74.2%

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

   if 2.0000000000000001e-61 < (*.f64 y y)

   1. Initial program 88.2%

      \[x \cdot x - y \cdot y \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 51.8%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-prod 11.9%

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

      7. add-sqr-sqrt 26.8%

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

   4. Applied egg-rr 26.8%

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

      1. add-sqr-sqrt 11.9%

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

      2. sqrt-prod 61.5%

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

      3. add-sqr-sqrt 11.9%

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

      4. add-sqr-sqrt 11.9%

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

      5. sqr-neg 11.9%

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

      6. swap-sqr 11.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 47.9%

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

      9. distribute-rgt-neg-out 47.9%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 78.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-185}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-61}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{-72} \lor \neg \left(x \cdot x \leq 1.5 \cdot 10^{+61}\right) \land x \cdot x \leq 10^{+167}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 6e-72)
         (and (not (<= (* x x) 1.5e+61)) (<= (* x x) 1e+167)))
   (* y (- y))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 6e-72) || (!((x * x) <= 1.5e+61) && ((x * x) <= 1e+167))) {
		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 * x) <= 6d-72) .or. (.not. ((x * x) <= 1.5d+61)) .and. ((x * x) <= 1d+167)) 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 * x) <= 6e-72) || (!((x * x) <= 1.5e+61) && ((x * x) <= 1e+167))) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 6e-72) or (not ((x * x) <= 1.5e+61) and ((x * x) <= 1e+167)):
		tmp = y * -y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 6e-72) || (!(Float64(x * x) <= 1.5e+61) && (Float64(x * x) <= 1e+167)))
		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 * x) <= 6e-72) || (~(((x * x) <= 1.5e+61)) && ((x * x) <= 1e+167)))
		tmp = y * -y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 6e-72], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.5e+61]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1e+167]]], N[(y * (-y)), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6e-72 or 1.5e61 < (*.f64 x x) < 1e167

   1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 46.1%

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

      4. sqrt-unprod 65.1%

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

      5. sqr-neg 65.1%

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

      6. sqrt-prod 18.8%

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

      7. add-sqr-sqrt 29.9%

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

   4. Applied egg-rr 29.9%

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

      1. add-sqr-sqrt 18.8%

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

      2. sqrt-prod 65.1%

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

      3. add-sqr-sqrt 18.8%

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

      4. add-sqr-sqrt 18.8%

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

      5. sqr-neg 18.8%

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

      6. swap-sqr 18.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 53.5%

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

      9. distribute-rgt-neg-out 53.5%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 83.2%

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

   8. Taylor expanded in x around 0 83.7%

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

      1. neg-mul-1 83.7%

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

   10. Simplified 83.7%

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

   if 6e-72 < (*.f64 x x) < 1.5e61 or 1e167 < (*.f64 x x)

   1. Initial program 87.7%

      \[x \cdot x - y \cdot y \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 50.8%

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

      4. sqrt-unprod 86.9%

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

      5. sqr-neg 86.9%

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

      6. sqrt-prod 38.5%

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

      7. add-sqr-sqrt 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in x around inf 89.2%

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

   6. Taylor expanded in x around inf 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{-72} \lor \neg \left(x \cdot x \leq 1.5 \cdot 10^{+61}\right) \land x \cdot x \leq 10^{+167}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% 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 94.1%

   \[x \cdot x - y \cdot y \]
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. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

5. Taylor expanded in x around inf 59.1%

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

6. Taylor expanded in x around inf 56.1%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))