Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 5

Speedup: 1.0×

• 43.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(x - y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e-62) (* x (+ x y)) (* y (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-62) {
		tmp = x * (x + y);
	} else {
		tmp = y * (x - 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 <= 2.6d-62) then
        tmp = x * (x + y)
    else
        tmp = y * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-62) {
		tmp = x * (x + y);
	} else {
		tmp = y * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e-62:
		tmp = x * (x + y)
	else:
		tmp = y * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-62)
		tmp = Float64(x * Float64(x + y));
	else
		tmp = Float64(y * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e-62)
		tmp = x * (x + y);
	else
		tmp = y * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-62], N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5999999999999999e-62

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 69.0%

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

   if 2.5999999999999999e-62 < y

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-61) (* x (+ x y)) (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-61) {
		tmp = x * (x + 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 <= 8.5d-61) then
        tmp = x * (x + y)
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-61) {
		tmp = x * (x + y);
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-61:
		tmp = x * (x + y)
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-61)
		tmp = Float64(x * Float64(x + y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-61)
		tmp = x * (x + y);
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-61], N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000016e-61

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 69.2%

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

   if 8.50000000000000016e-61 < y

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.1%

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

   6. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot y \]
   7. Step-by-step derivation

      1. neg-mul-1 76.2%

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

   8. Simplified 76.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+221}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.1e+221) (* y (- y)) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+221) {
		tmp = y * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.1d+221) then
        tmp = y * -y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+221) {
		tmp = y * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.1e+221:
		tmp = y * -y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.1e+221)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.1e+221)
		tmp = y * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e+221], N[(y * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000002e221

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 59.6%

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

   6. Taylor expanded in x around 0 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot y\right)} \cdot y \]
   7. Step-by-step derivation

      1. neg-mul-1 58.5%

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

   8. Simplified 58.5%

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

   if 2.10000000000000002e221 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(x - y\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in x around 0 34.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+221}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 15.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(x - y\right) \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 57.0%

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

6. Taylor expanded in x around 0 16.2%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))