Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.1% → 97.3%

Time: 2.3s

Alternatives: 4

Speedup: 0.6×

• 43.6% 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: 93.1% 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: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+234}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+234) (* x x) (fma x x (* y (- y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+234) {
		tmp = x * x;
	} else {
		tmp = fma(x, x, (y * -y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+234)
		tmp = Float64(x * x);
	else
		tmp = fma(x, x, Float64(y * Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e+234], N[(x * x), $MachinePrecision], N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.40000000000000011e234

   1. Initial program 66.7%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if -2.40000000000000011e234 < x

   1. Initial program 93.4%

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

      1. fma-neg 97.5%

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

      2. distribute-rgt-neg-in 97.5%

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

   3. Simplified 97.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+234}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 2: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.4 \cdot 10^{-36} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+63}\right) \land x \cdot x \leq 5.5 \cdot 10^{+144}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 7.4e-36)
         (and (not (<= (* x x) 2e+63)) (<= (* x x) 5.5e+144)))
   (* y (- y))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 7.4e-36) || (!((x * x) <= 2e+63) && ((x * x) <= 5.5e+144))) {
		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) <= 7.4d-36) .or. (.not. ((x * x) <= 2d+63)) .and. ((x * x) <= 5.5d+144)) 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) <= 7.4e-36) || (!((x * x) <= 2e+63) && ((x * x) <= 5.5e+144))) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 7.4e-36) or (not ((x * x) <= 2e+63) and ((x * x) <= 5.5e+144)):
		tmp = y * -y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 7.4e-36) || (!(Float64(x * x) <= 2e+63) && (Float64(x * x) <= 5.5e+144)))
		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) <= 7.4e-36) || (~(((x * x) <= 2e+63)) && ((x * x) <= 5.5e+144)))
		tmp = y * -y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 7.4e-36], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 2e+63]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 5.5e+144]]], N[(y * (-y)), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 7.40000000000000003e-36 or 2.00000000000000012e63 < (*.f64 x x) < 5.50000000000000022e144

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.9%

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

      1. unpow2 84.9%

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

      2. mul-1-neg 84.9%

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

      3. distribute-rgt-neg-in 84.9%

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

   4. Simplified 84.9%

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

   if 7.40000000000000003e-36 < (*.f64 x x) < 2.00000000000000012e63 or 5.50000000000000022e144 < (*.f64 x x)

   1. Initial program 82.8%

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

   2. Taylor expanded in x around inf 78.2%

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

      1. unpow2 78.3%

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

   4. Simplified 78.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.4 \cdot 10^{-36} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+63}\right) \land x \cdot x \leq 5.5 \cdot 10^{+144}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 3e+294) (- (* x x) (* y y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3e+294) {
		tmp = (x * x) - (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) <= 3d+294) then
        tmp = (x * x) - (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) <= 3e+294) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 3e+294:
		tmp = (x * x) - (y * y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 3e+294)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 3e+294)
		tmp = (x * x) - (y * y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 3e+294], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.00000000000000006e294

   1. Initial program 100.0%

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

   if 3.00000000000000006e294 < (*.f64 x x)

   1. Initial program 68.2%

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

   2. Taylor expanded in x around inf 84.8%

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

      1. unpow2 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 53.9% 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 91.8%

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

2. Taylor expanded in x around inf 51.4%

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

   1. unpow2 51.4%

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

4. Simplified 51.4%

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

5. Final simplification 51.4%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))