Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.0% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 0.6×

• 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: 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: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \mathsf{fma}\left(x, \frac{1}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+307)
   (- (* x x) (* y y))
   (* (pow y 2.0) (fma x (* (/ 1.0 y) (/ x y)) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = pow(y, 2.0) * fma(x, ((1.0 / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+307)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64((y ^ 2.0) * fma(x, Float64(Float64(1.0 / y) * Float64(x / y)), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+307], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * N[(x * N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 y y)

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{{x}^{2}}{{y}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {y}^{2} \cdot \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {y}^{2} \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{\color{blue}{1 \cdot x}}{{y}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{1 \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(-1 + {\left(\frac{x}{y}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+307)
   (- (* x x) (* y y))
   (* (pow y 2.0) (+ -1.0 (pow (/ x y) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = pow(y, 2.0) * (-1.0 + pow((x / y), 2.0));
	}
	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) <= 5d+307) then
        tmp = (x * x) - (y * y)
    else
        tmp = (y ** 2.0d0) * ((-1.0d0) + ((x / y) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = Math.pow(y, 2.0) * (-1.0 + Math.pow((x / y), 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+307:
		tmp = (x * x) - (y * y)
	else:
		tmp = math.pow(y, 2.0) * (-1.0 + math.pow((x / y), 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+307)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64((y ^ 2.0) * Float64(-1.0 + (Float64(x / y) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+307)
		tmp = (x * x) - (y * y);
	else
		tmp = (y ^ 2.0) * (-1.0 + ((x / y) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+307], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * N[(-1.0 + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 y y)

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{{x}^{2}}{{y}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {y}^{2} \cdot \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {y}^{2} \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{\color{blue}{1 \cdot x}}{{y}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{1 \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]
   8. Step-by-step derivation

      1. fma-undefine 100.0%

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{y} \cdot \frac{x}{y}\right) + -1\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{\left(x \cdot \frac{1}{y}\right) \cdot \frac{x}{y}} + -1\right) \]

      3. div-inv 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + -1\right) \]

      4. pow2 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + -1\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto {y}^{2} \cdot \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(-1 + {\left(\frac{x}{y}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(-1 + \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+307)
   (- (* x x) (* y y))
   (* (pow y 2.0) (+ -1.0 (/ (/ x y) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = pow(y, 2.0) * (-1.0 + ((x / y) / (y / x)));
	}
	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) <= 5d+307) then
        tmp = (x * x) - (y * y)
    else
        tmp = (y ** 2.0d0) * ((-1.0d0) + ((x / y) / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = Math.pow(y, 2.0) * (-1.0 + ((x / y) / (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+307:
		tmp = (x * x) - (y * y)
	else:
		tmp = math.pow(y, 2.0) * (-1.0 + ((x / y) / (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+307)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64((y ^ 2.0) * Float64(-1.0 + Float64(Float64(x / y) / Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+307)
		tmp = (x * x) - (y * y);
	else
		tmp = (y ^ 2.0) * (-1.0 + ((x / y) / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+307], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * N[(-1.0 + N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 y y)

   1. Initial program 71.7%

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

   3. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{{x}^{2}}{{y}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {y}^{2} \cdot \left(\frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{x \cdot \frac{x}{{y}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {y}^{2} \cdot \color{blue}{\mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \mathsf{fma}\left(x, \frac{x}{{y}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{\color{blue}{1 \cdot x}}{{y}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \frac{1 \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {y}^{2} \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{y} \cdot \frac{x}{y}}, -1\right) \]
   8. Step-by-step derivation

      1. fma-undefine 100.0%

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{y} \cdot \frac{x}{y}\right) + -1\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{\left(x \cdot \frac{1}{y}\right) \cdot \frac{x}{y}} + -1\right) \]

      3. div-inv 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + -1\right) \]

      4. pow2 100.0%

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + -1\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto {y}^{2} \cdot \color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + -1\right)} \]
   10. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + -1\right) \]

       2. clear-num 100.0%

          \[\leadsto {y}^{2} \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + -1\right) \]

       3. un-div-inv 100.0%

          \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1\right) \]

   11. Applied egg-rr 100.0%

       \[\leadsto {y}^{2} \cdot \left(\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(-1 + \frac{\frac{x}{y}}{\frac{y}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e+208) (fma x x (* y (- y))) (* (+ y x) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+208) {
		tmp = fma(x, x, (y * -y));
	} else {
		tmp = (y + x) * (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e+208)
		tmp = fma(x, x, Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(y + x) * Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e+208], N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000001e208

   1. Initial program 94.5%

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

      1. sqr-neg 94.5%

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

      2. cancel-sign-sub 94.5%

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

      3. fma-define 96.6%

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

   3. Simplified 96.6%

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

   if 3.2000000000000001e208 < x

   1. Initial program 77.8%

      \[x \cdot x - y \cdot y \]
   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 44.4%

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

      4. sqrt-unprod 94.4%

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

      5. sqr-neg 94.4%

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

      6. sqrt-prod 50.0%

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

      7. add-sqr-sqrt 94.4%

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

   4. Applied egg-rr 94.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.5e+161) (- (* x x) (* y y)) (* (+ y x) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.5e+161) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = (y + x) * (y + 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 <= 7.5d+161) then
        tmp = (x * x) - (y * y)
    else
        tmp = (y + x) * (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.5e+161) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = (y + x) * (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.5e+161:
		tmp = (x * x) - (y * y)
	else:
		tmp = (y + x) * (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.5e+161)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(Float64(y + x) * Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.5e+161)
		tmp = (x * x) - (y * y);
	else
		tmp = (y + x) * (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.5e+161], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999995e161

   1. Initial program 94.4%

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

   if 7.4999999999999995e161 < x

   1. Initial program 82.6%

      \[x \cdot x - y \cdot y \]
   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 43.5%

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

      4. sqrt-unprod 95.7%

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

      5. sqr-neg 95.7%

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

      6. sqrt-prod 52.2%

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

      7. add-sqr-sqrt 95.7%

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

   4. Applied egg-rr 95.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + x\right) \cdot \left(y + x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

   \[x \cdot x - y \cdot y \]
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 49.5%

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

   4. sqrt-unprod 71.9%

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

   5. sqr-neg 71.9%

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

   6. sqrt-prod 25.4%

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

   7. add-sqr-sqrt 54.3%

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

4. Applied egg-rr 54.3%

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

5. Final simplification 54.3%

   \[\leadsto \left(y + x\right) \cdot \left(y + x\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))