Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 5

Speedup: 1.0×

• 44.4% 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: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.1e+42) (* y (- x y)) (* x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.1e+42) {
		tmp = y * (x - y);
	} else {
		tmp = x * (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 <= 5.1d+42) then
        tmp = y * (x - y)
    else
        tmp = x * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.1e+42) {
		tmp = y * (x - y);
	} else {
		tmp = x * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.1e+42:
		tmp = y * (x - y)
	else:
		tmp = x * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.1e+42)
		tmp = Float64(y * Float64(x - y));
	else
		tmp = Float64(x * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.1e+42)
		tmp = y * (x - y);
	else
		tmp = x * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.1e+42], N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0999999999999999e42

   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 67.2%

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

   if 5.0999999999999999e42 < 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 88.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+43}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+43) (* y (- y)) (* x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+43) {
		tmp = y * -y;
	} else {
		tmp = x * (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 <= 1d+43) then
        tmp = y * -y
    else
        tmp = x * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+43) {
		tmp = y * -y;
	} else {
		tmp = x * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+43:
		tmp = y * -y
	else:
		tmp = x * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+43)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+43)
		tmp = y * -y;
	else
		tmp = x * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+43], N[(y * (-y)), $MachinePrecision], N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000001e43

   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 67.5%

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

      1. neg-mul-1 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in x around 0 66.3%

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

   if 1.00000000000000001e43 < 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 88.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+43}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.051:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.051) (* x x) (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.051) {
		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 <= 0.051d0) 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 <= 0.051) {
		tmp = x * x;
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.051:
		tmp = x * x
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.051)
		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 <= 0.051)
		tmp = x * x;
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.051], N[(x * x), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0509999999999999967

   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 63.0%

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

      1. *-commutative 63.0%

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

      2. sub-neg 63.0%

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

      3. distribute-rgt-in 61.9%

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

      4. pow2 61.9%

         \[\leadsto \color{blue}{{x}^{2}} + \left(-y\right) \cdot x \]

      5. add-sqr-sqrt 32.3%

         \[\leadsto {x}^{2} + \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \cdot x \]

      6. sqrt-unprod 66.4%

         \[\leadsto {x}^{2} + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}} \cdot x \]

      7. sqr-neg 66.4%

         \[\leadsto {x}^{2} + \sqrt{\color{blue}{y \cdot y}} \cdot x \]

      8. sqrt-unprod 29.6%

         \[\leadsto {x}^{2} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot x \]

      9. add-sqr-sqrt 62.0%

         \[\leadsto {x}^{2} + \color{blue}{y} \cdot x \]

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{{x}^{2} + y \cdot x} \]
   8. Step-by-step derivation

      1. unpow2 62.0%

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

      2. distribute-rgt-in 63.6%

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

   9. Simplified 63.6%

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

   10. Taylor expanded in x around inf 61.2%

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

   if 0.0509999999999999967 < 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 83.7%

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

      1. neg-mul-1 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in x around 0 80.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.051:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% 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 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 54.6%

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

   1. *-commutative 54.6%

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

   2. sub-neg 54.6%

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

   3. distribute-rgt-in 51.4%

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

   4. pow2 51.4%

      \[\leadsto \color{blue}{{x}^{2}} + \left(-y\right) \cdot x \]

   5. add-sqr-sqrt 24.1%

      \[\leadsto {x}^{2} + \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \cdot x \]

   6. sqrt-unprod 58.0%

      \[\leadsto {x}^{2} + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}} \cdot x \]

   7. sqr-neg 58.0%

      \[\leadsto {x}^{2} + \sqrt{\color{blue}{y \cdot y}} \cdot x \]

   8. sqrt-unprod 27.2%

      \[\leadsto {x}^{2} + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot x \]

   9. add-sqr-sqrt 51.3%

      \[\leadsto {x}^{2} + \color{blue}{y} \cdot x \]

7. Applied egg-rr 51.3%

   \[\leadsto \color{blue}{{x}^{2} + y \cdot x} \]
8. Step-by-step derivation

   1. unpow2 51.3%

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

   2. distribute-rgt-in 54.5%

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

9. Simplified 54.5%

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

10. Taylor expanded in x around inf 51.0%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))