Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.1% → 97.0%

Time: 2.2s

Alternatives: 4

Speedup: 0.6×

• 43.0% 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: 94.1% 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: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. fma-neg 97.7%

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

   2. distribute-rgt-neg-in 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 2: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+306) (- (* x x) (* y y)) (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+306) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * -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+306) then
        tmp = (x * x) - (y * y)
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+306) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+306:
		tmp = (x * x) - (y * y)
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+306)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e+306)
		tmp = (x * x) - (y * y);
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+306], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000003e306

   1. Initial program 100.0%

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

   if 2.00000000000000003e306 < (*.f64 y y)

   1. Initial program 77.4%

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

   2. Taylor expanded in x around 0 90.3%

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

      1. unpow2 90.3%

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

      2. mul-1-neg 90.3%

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

      3. distribute-rgt-neg-in 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]

Alternative 3: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-66} \lor \neg \left(y \leq 2.65 \cdot 10^{-39}\right) \land y \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 7.5e-66) (and (not (<= y 2.65e-39)) (<= y 2.4e+48)))
   (* x x)
   (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 7.5e-66) || (!(y <= 2.65e-39) && (y <= 2.4e+48))) {
		tmp = x * x;
	} else {
		tmp = y * -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 <= 7.5d-66) .or. (.not. (y <= 2.65d-39)) .and. (y <= 2.4d+48)) then
        tmp = x * x
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 7.5e-66) || (!(y <= 2.65e-39) && (y <= 2.4e+48))) {
		tmp = x * x;
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 7.5e-66) or (not (y <= 2.65e-39) and (y <= 2.4e+48)):
		tmp = x * x
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 7.5e-66) || (!(y <= 2.65e-39) && (y <= 2.4e+48)))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 7.5e-66) || (~((y <= 2.65e-39)) && (y <= 2.4e+48)))
		tmp = x * x;
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 7.5e-66], And[N[Not[LessEqual[y, 2.65e-39]], $MachinePrecision], LessEqual[y, 2.4e+48]]], N[(x * x), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999995e-66 or 2.65000000000000002e-39 < y < 2.4000000000000001e48

   1. Initial program 96.9%

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

   2. Taylor expanded in x around inf 66.5%

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

      1. unpow2 66.5%

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

   4. Simplified 66.5%

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

   if 7.49999999999999995e-66 < y < 2.65000000000000002e-39 or 2.4000000000000001e48 < y

   1. Initial program 87.1%

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

   2. Taylor expanded in x around 0 82.3%

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

      1. unpow2 82.3%

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

      2. mul-1-neg 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

   4. Simplified 82.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-66} \lor \neg \left(y \leq 2.65 \cdot 10^{-39}\right) \land y \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]

Alternative 4: 54.2% 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 94.5%

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

2. Taylor expanded in x around inf 54.9%

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

   1. unpow2 54.9%

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

4. Simplified 54.9%

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

5. Final simplification 54.9%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))