Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.8% → 100.0%

Time: 5.7s

Alternatives: 3

Speedup: 1.2×

• 43.5% 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: 93.8% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\left(-y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (* x x) (* y y)) -2e-281) (* (- y) y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) - (y * y)) <= -2e-281) {
		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) - (y * y)) <= (-2d-281)) 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) - (y * y)) <= -2e-281) {
		tmp = -y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) - (y * y)) <= -2e-281:
		tmp = -y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) - Float64(y * y)) <= -2e-281)
		tmp = Float64(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) - (y * y)) <= -2e-281)
		tmp = -y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], -2e-281], N[((-y) * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 y y)) < -2e-281

   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

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -2e-281 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 84.7%

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

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 19.7

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

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.6

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

   8. Applied rewrites 92.6%

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

Alternative 3: 52.9% 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 91.4%

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

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

   2. unpow2 N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 52.9

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

8. Applied rewrites 52.9%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))