Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.7% → 100.0%

Time: 4.7s

Alternatives: 7

Speedup: 0.4×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. +-commutative 91.0%

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

   2. associate-+l+ 91.0%

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

   3. +-commutative 91.0%

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

   4. associate-+l+ 91.0%

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

   5. distribute-rgt-out 95.7%

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

   6. fma-define 100.0%

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

   7. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, y\right), x \cdot x\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(x, 2, y\right), x \cdot x\right) \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y y) (+ (* x x) (* y (* x 2.0))))))
   (if (<= t_0 INFINITY) t_0 (pow y 2.0))))

C

double code(double x, double y) {
	double t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = pow(y, 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = Math.pow(y, 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) + ((x * x) + (y * (x * 2.0)))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = math.pow(y, 2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) + Float64(Float64(x * x) + Float64(y * Float64(x * 2.0))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = y ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = y ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[Power[y, 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 72.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ {x}^{2} + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around inf 99.0%

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

4. Final simplification 99.0%

   \[\leadsto {x}^{2} + y \cdot y \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y y) (+ (* x x) (* y (* x 2.0))))))
   (if (<= t_0 INFINITY) t_0 (* x (+ x (* y 2.0))))))

C

double code(double x, double y) {
	double t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = x * (x + (y * 2.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = x * (x + (y * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) + ((x * x) + (y * (x * 2.0)))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = x * (x + (y * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) + Float64(Float64(x * x) + Float64(y * Float64(x * 2.0))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) + ((x * x) + (y * (x * 2.0)));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = x * (x + (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 0.0%

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

      1. associate-+l+ 0.0%

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

      2. fma-define 0.0%

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

      3. distribute-rgt-in 52.2%

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

      4. fma-undefine 52.2%

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

   4. Applied egg-rr 52.2%

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

   5. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*r* 0.0%

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

      5. distribute-rgt-out 47.8%

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

      6. *-commutative 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq \infty:\\ \;\;\;\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -55000 \lor \neg \left(x \leq -7.5 \cdot 10^{-25}\right) \land x \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot \left(y \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -55000.0) (and (not (<= x -7.5e-25)) (<= x -6.5e-81)))
   (* x (+ x (* y 2.0)))
   (+ (* y y) (* x (* y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -55000.0) || (!(x <= -7.5e-25) && (x <= -6.5e-81))) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = (y * y) + (x * (y * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-55000.0d0)) .or. (.not. (x <= (-7.5d-25))) .and. (x <= (-6.5d-81))) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = (y * y) + (x * (y * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -55000.0) || (!(x <= -7.5e-25) && (x <= -6.5e-81))) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = (y * y) + (x * (y * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -55000.0) or (not (x <= -7.5e-25) and (x <= -6.5e-81)):
		tmp = x * (x + (y * 2.0))
	else:
		tmp = (y * y) + (x * (y * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -55000.0) || (!(x <= -7.5e-25) && (x <= -6.5e-81)))
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(y * y) + Float64(x * Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -55000.0) || (~((x <= -7.5e-25)) && (x <= -6.5e-81)))
		tmp = x * (x + (y * 2.0));
	else
		tmp = (y * y) + (x * (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -55000.0], And[N[Not[LessEqual[x, -7.5e-25]], $MachinePrecision], LessEqual[x, -6.5e-81]]], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -55000 or -7.49999999999999989e-25 < x < -6.5000000000000002e-81

   1. Initial program 84.7%

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

      1. associate-+l+ 84.7%

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

      2. fma-define 84.7%

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

      3. distribute-rgt-in 91.8%

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

      4. fma-undefine 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in x around inf 73.4%

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

      1. +-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*r* 73.4%

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

      5. distribute-rgt-out 81.6%

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

      6. *-commutative 81.6%

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

   7. Simplified 81.6%

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

   if -55000 < x < -7.49999999999999989e-25 or -6.5000000000000002e-81 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 57.7%

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

      1. associate-*r* 57.7%

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

      2. *-commutative 57.7%

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

      3. associate-*r* 57.7%

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

   5. Simplified 57.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -55000 \lor \neg \left(x \leq -7.5 \cdot 10^{-25}\right) \land x \leq -6.5 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot \left(y \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. associate-+l+ 91.0%

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

   2. fma-define 91.0%

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

   3. distribute-rgt-in 95.7%

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

   4. fma-undefine 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in x around inf 56.7%

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

   1. +-commutative 56.7%

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

   2. unpow2 56.7%

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

   3. *-commutative 56.7%

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

   4. associate-*r* 56.7%

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

   5. distribute-rgt-out 61.0%

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

   6. *-commutative 61.0%

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

7. Simplified 61.0%

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

8. Final simplification 61.0%

   \[\leadsto x \cdot \left(x + y \cdot 2\right) \]
9. Add Preprocessing

Alternative 7: 15.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around 0 46.7%

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

   1. associate-*r* 46.7%

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

   2. *-commutative 46.7%

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

   3. associate-*r* 46.7%

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

5. Simplified 46.7%

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

6. Taylor expanded in x around inf 12.9%

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

   1. *-commutative 12.9%

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

8. Simplified 12.9%

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

9. Final simplification 12.9%

   \[\leadsto 2 \cdot \left(y \cdot x\right) \]
10. Add Preprocessing

Developer target: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))