Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 3

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. Final simplification 100.0%

   \[\leadsto \left(x + y\right) \cdot \left(x - y\right) \]

Alternative 2: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.4e+18) (* y (- y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4e+18) {
		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 <= 1.4d+18) 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 <= 1.4e+18) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4e+18:
		tmp = y * -y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4e+18)
		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 <= 1.4e+18)
		tmp = y * -y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4e+18], N[(y * (-y)), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e18

   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. Taylor expanded in x around 0 61.1%

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

      1. unpow2 61.1%

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

      2. neg-mul-1 61.1%

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

      3. distribute-rgt-neg-in 61.1%

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

   6. Simplified 61.1%

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

   if 1.4e18 < 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. Taylor expanded in x around inf 78.2%

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

      1. unpow2 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 3: 53.4% 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. Taylor expanded in x around inf 54.6%

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

   1. unpow2 54.6%

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

6. Simplified 54.6%

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

7. Final simplification 54.6%

   \[\leadsto x \cdot x \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))