Examples.Basics.BasicTests:f3 from sbv-4.4

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 4

Speedup: 1.0×

• 43.9% 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: 67.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.076:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-88}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.076)
   (* x x)
   (if (<= x -2.9e-88)
     (* y y)
     (if (<= x -5.8e-141) (* x (+ x (* y 2.0))) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.076) {
		tmp = x * x;
	} else if (x <= -2.9e-88) {
		tmp = y * y;
	} else if (x <= -5.8e-141) {
		tmp = x * (x + (y * 2.0));
	} 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 (x <= (-0.076d0)) then
        tmp = x * x
    else if (x <= (-2.9d-88)) then
        tmp = y * y
    else if (x <= (-5.8d-141)) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.076) {
		tmp = x * x;
	} else if (x <= -2.9e-88) {
		tmp = y * y;
	} else if (x <= -5.8e-141) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.076:
		tmp = x * x
	elif x <= -2.9e-88:
		tmp = y * y
	elif x <= -5.8e-141:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.076)
		tmp = Float64(x * x);
	elseif (x <= -2.9e-88)
		tmp = Float64(y * y);
	elseif (x <= -5.8e-141)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.076)
		tmp = x * x;
	elseif (x <= -2.9e-88)
		tmp = y * y;
	elseif (x <= -5.8e-141)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.076], N[(x * x), $MachinePrecision], If[LessEqual[x, -2.9e-88], N[(y * y), $MachinePrecision], If[LessEqual[x, -5.8e-141], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0759999999999999981

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 84.8%

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

      1. unpow2 84.8%

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

   4. Simplified 84.8%

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

   if -0.0759999999999999981 < x < -2.9000000000000001e-88 or -5.7999999999999999e-141 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 72.0%

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

      1. unpow2 72.0%

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

   4. Simplified 72.0%

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

   if -2.9000000000000001e-88 < x < -5.7999999999999999e-141

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 56.4%

      \[\leadsto \color{blue}{2 \cdot \left(y \cdot x\right) + {x}^{2}} \]
   3. Step-by-step derivation

      1. associate-*r* 56.4%

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

      2. unpow2 56.4%

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

      3. distribute-rgt-out 56.4%

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

      4. *-commutative 56.4%

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

   4. Simplified 56.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.076:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-88}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 3: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-9} \lor \neg \left(x \leq -2.9 \cdot 10^{-88}\right) \land x \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.9e-9) (and (not (<= x -2.9e-88)) (<= x -5.2e-141)))
   (* x x)
   (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.9e-9) || (!(x <= -2.9e-88) && (x <= -5.2e-141))) {
		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 ((x <= (-3.9d-9)) .or. (.not. (x <= (-2.9d-88))) .and. (x <= (-5.2d-141))) 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 ((x <= -3.9e-9) || (!(x <= -2.9e-88) && (x <= -5.2e-141))) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.9e-9) or (not (x <= -2.9e-88) and (x <= -5.2e-141)):
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.9e-9) || (!(x <= -2.9e-88) && (x <= -5.2e-141)))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.9e-9) || (~((x <= -2.9e-88)) && (x <= -5.2e-141)))
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.9e-9], And[N[Not[LessEqual[x, -2.9e-88]], $MachinePrecision], LessEqual[x, -5.2e-141]]], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000002e-9 or -2.9000000000000001e-88 < x < -5.20000000000000022e-141

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 80.6%

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

      1. unpow2 80.6%

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

   4. Simplified 80.6%

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

   if -3.9000000000000002e-9 < x < -2.9000000000000001e-88 or -5.20000000000000022e-141 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 72.0%

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

      1. unpow2 72.0%

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

   4. Simplified 72.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-9} \lor \neg \left(x \leq -2.9 \cdot 10^{-88}\right) \land x \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4: 56.3% 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. Taylor expanded in x around inf 56.2%

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

   1. unpow2 56.2%

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

4. Simplified 56.2%

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

5. Final simplification 56.2%

   \[\leadsto x \cdot x \]

Developer target: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + 2 \cdot \left(y \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (+ (* y y) (* 2.0 (* y x)))))

C

double code(double x, double y) {
	return (x * x) + ((y * y) + (2.0 * (y * x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) + (2.0d0 * (y * x)))
end function

Java

public static double code(double x, double y) {
	return (x * x) + ((y * y) + (2.0 * (y * x)));
}

Python

def code(x, y):
	return (x * x) + ((y * y) + (2.0 * (y * x)))

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(2.0 * Float64(y * x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + (2.0 * (y * x)));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(2.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* 2.0 (* y x))))

  (* (+ x y) (+ x y)))