Examples.Basics.BasicTests:f1 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 5

Speedup: 1.0×

• 44.8% 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: 66.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-36} \lor \neg \left(x \leq 4.4 \cdot 10^{+30}\right) \land x \leq 1.66 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 2.45e-36) (and (not (<= x 4.4e+30)) (<= x 1.66e+52)))
   (* y (- y))
   (* x (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 2.45e-36) || (!(x <= 4.4e+30) && (x <= 1.66e+52))) {
		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 <= 2.45d-36) .or. (.not. (x <= 4.4d+30)) .and. (x <= 1.66d+52)) 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 <= 2.45e-36) || (!(x <= 4.4e+30) && (x <= 1.66e+52))) {
		tmp = y * -y;
	} else {
		tmp = x * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 2.45e-36) or (not (x <= 4.4e+30) and (x <= 1.66e+52)):
		tmp = y * -y
	else:
		tmp = x * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 2.45e-36) || (!(x <= 4.4e+30) && (x <= 1.66e+52)))
		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 <= 2.45e-36) || (~((x <= 4.4e+30)) && (x <= 1.66e+52)))
		tmp = y * -y;
	else
		tmp = x * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 2.45e-36], And[N[Not[LessEqual[x, 4.4e+30]], $MachinePrecision], LessEqual[x, 1.66e+52]]], N[(y * (-y)), $MachinePrecision], N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4499999999999998e-36 or 4.4e30 < x < 1.65999999999999994e52

   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 67.3%

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

   6. Taylor expanded in x around 0 65.7%

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

      1. neg-mul-1 65.7%

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

   8. Simplified 65.7%

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

   if 2.4499999999999998e-36 < x < 4.4e30 or 1.65999999999999994e52 < 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 85.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-36} \lor \neg \left(x \leq 4.4 \cdot 10^{+30}\right) \land x \leq 1.66 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x - y\right)\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- x y))))
   (if (<= y 4.8e-93)
     t_0
     (if (<= y 5.5e-67) (* y (- y)) (if (<= y 4.6e-30) t_0 (* y (- x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (x - y);
	double tmp;
	if (y <= 4.8e-93) {
		tmp = t_0;
	} else if (y <= 5.5e-67) {
		tmp = y * -y;
	} else if (y <= 4.6e-30) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (x - y)
    if (y <= 4.8d-93) then
        tmp = t_0
    else if (y <= 5.5d-67) then
        tmp = y * -y
    else if (y <= 4.6d-30) then
        tmp = t_0
    else
        tmp = y * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x - y);
	double tmp;
	if (y <= 4.8e-93) {
		tmp = t_0;
	} else if (y <= 5.5e-67) {
		tmp = y * -y;
	} else if (y <= 4.6e-30) {
		tmp = t_0;
	} else {
		tmp = y * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x - y)
	tmp = 0
	if y <= 4.8e-93:
		tmp = t_0
	elif y <= 5.5e-67:
		tmp = y * -y
	elif y <= 4.6e-30:
		tmp = t_0
	else:
		tmp = y * (x - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x - y))
	tmp = 0.0
	if (y <= 4.8e-93)
		tmp = t_0;
	elseif (y <= 5.5e-67)
		tmp = Float64(y * Float64(-y));
	elseif (y <= 4.6e-30)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x - y);
	tmp = 0.0;
	if (y <= 4.8e-93)
		tmp = t_0;
	elseif (y <= 5.5e-67)
		tmp = y * -y;
	elseif (y <= 4.6e-30)
		tmp = t_0;
	else
		tmp = y * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-93], t$95$0, If[LessEqual[y, 5.5e-67], N[(y * (-y)), $MachinePrecision], If[LessEqual[y, 4.6e-30], t$95$0, N[(y * N[(x - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.8000000000000002e-93 or 5.5000000000000003e-67 < y < 4.59999999999999968e-30

   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 67.8%

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

   if 4.8000000000000002e-93 < y < 5.5000000000000003e-67

   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 72.9%

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

   6. Taylor expanded in x around 0 76.1%

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

      1. neg-mul-1 76.1%

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

   8. Simplified 76.1%

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

   if 4.59999999999999968e-30 < 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 81.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+172}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.6e+172) (* y (- y)) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.6e+172) {
		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 <= 2.6d+172) 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 <= 2.6e+172) {
		tmp = y * -y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.6e+172:
		tmp = y * -y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.6e+172)
		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 <= 2.6e+172)
		tmp = y * -y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.6e+172], N[(y * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e172

   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 61.3%

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

   6. Taylor expanded in x around 0 60.0%

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

      1. neg-mul-1 60.0%

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

   8. Simplified 60.0%

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

   if 2.6e172 < 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 0 19.3%

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

   6. Taylor expanded in x around inf 14.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+172}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 14.9% 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 0 54.9%

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

6. Taylor expanded in x around inf 14.9%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f1 from sbv-4.4"
  :precision binary64
  (* (+ x y) (- x y)))