Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.0% → 96.7%

Time: 2.5s

Alternatives: 3

Speedup: 0.5×

• 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: 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: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.2 \cdot 10^{+302}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2.2e+302) (- (* x x) (* y y)) (* y (- 0.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.2e+302) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * (0.0 - 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) <= 2.2d+302) then
        tmp = (x * x) - (y * y)
    else
        tmp = y * (0.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.2e+302) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * (0.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2.2e+302:
		tmp = (x * x) - (y * y)
	else:
		tmp = y * (0.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2.2e+302)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(y * Float64(0.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2.2e+302)
		tmp = (x * x) - (y * y);
	else
		tmp = y * (0.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2.2e+302], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.2000000000000001e302

   1. Initial program 100.0%

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

   if 2.2000000000000001e302 < (*.f64 y y)

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{y}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 93.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 93.7%

      \[\leadsto \color{blue}{0 - y \cdot y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot y\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot y\right)\right) \]

      3. *-lowering-*.f64 93.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, y\right)\right) \]

   7. Applied egg-rr 93.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.2 \cdot 10^{+302}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4.2e-88) (* y (- 0.0 y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4.2e-88) {
		tmp = y * (0.0 - 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) <= 4.2d-88) then
        tmp = y * (0.0d0 - 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) <= 4.2e-88) {
		tmp = y * (0.0 - y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 4.2e-88:
		tmp = y * (0.0 - y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4.2e-88)
		tmp = Float64(y * Float64(0.0 - y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 4.2e-88)
		tmp = y * (0.0 - y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.2e-88], N[(y * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.1999999999999999e-88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{2}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{y}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{y}\right)\right) \]

      5. *-lowering-*.f64 86.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 86.8%

      \[\leadsto \color{blue}{0 - y \cdot y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot y\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot y\right)\right) \]

      3. *-lowering-*.f64 86.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, y\right)\right) \]

   7. Applied egg-rr 86.8%

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

   if 4.1999999999999999e-88 < (*.f64 x x)

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 73.9%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 73.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.0% 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 96.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 49.5%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

5. Simplified 49.5%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))