Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.6% → 100.0%

Time: 5.0s

Alternatives: 7

Speedup: 0.4×

• 43.0% 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: 93.6% 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 94.5%

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

   1. +-commutative 94.5%

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

   2. associate-+l+ 94.5%

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

   3. +-commutative 94.5%

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

   4. associate-+l+ 94.5%

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

   5. distribute-rgt-out 96.9%

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

   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: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[Power[x, 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. 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. associate-*l* 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. fma-define 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in x around inf 80.4%

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

4. Final simplification 98.9%

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

Alternative 3: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \mathsf{fma}\left(x, 2, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+188) (* x (+ x (* y 2.0))) (+ (* x x) (* y (fma x 2.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+188) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = (x * x) + (y * fma(x, 2.0, y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+188)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(x * x) + Float64(y * fma(x, 2.0, y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e+188], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000021e188

   1. Initial program 75.9%

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

      1. associate-+l+ 75.9%

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

      2. associate-*l* 75.9%

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

      3. *-commutative 75.9%

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

      4. *-commutative 75.9%

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

      5. +-commutative 75.9%

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

      6. fma-define 75.9%

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

      7. *-commutative 75.9%

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

      8. *-commutative 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in y around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 75.9%

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

      4. *-commutative 75.9%

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

   7. Simplified 75.9%

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

      1. +-commutative 75.9%

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

      2. associate-*r* 75.9%

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

      3. distribute-rgt-out 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -3.60000000000000021e188 < x

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-*l* 96.9%

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

      3. *-commutative 96.9%

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

      4. *-commutative 96.9%

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

      5. +-commutative 96.9%

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

      6. fma-define 96.9%

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

      7. *-commutative 96.9%

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

      8. *-commutative 96.9%

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

   3. Simplified 96.9%

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

      1. fma-undefine 96.9%

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

      2. associate-*r* 96.9%

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

      3. distribute-rgt-out 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-undefine 99.5%

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

      6. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \mathsf{fma}\left(x, 2, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. +-commutative 94.5%

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

   2. fma-define 94.5%

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

   3. associate-*l* 94.5%

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

   4. *-commutative 94.5%

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

   5. distribute-lft-out 97.7%

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

   6. *-commutative 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 5: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = ((x * x) + (y * (x * 2.0))) + (y * y);
	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 = ((x * x) + (y * (x * 2.0))) + (y * y);
	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 = ((x * x) + (y * (x * 2.0))) + (y * y)
	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(Float64(x * x) + Float64(y * Float64(x * 2.0))) + Float64(y * y))
	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 = ((x * x) + (y * (x * 2.0))) + (y * y);
	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[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $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. 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. associate-*l* 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. fma-define 0.0%

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

      7. *-commutative 0.0%

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

      8. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in y around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 57.1%

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

   9. Applied egg-rr 57.1%

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

4. Final simplification 97.7%

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

Alternative 6: 57.4% 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 94.5%

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

   1. associate-+l+ 94.5%

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

   2. associate-*l* 94.5%

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

   3. *-commutative 94.5%

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

   4. *-commutative 94.5%

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

   5. +-commutative 94.5%

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

   6. fma-define 94.5%

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

   7. *-commutative 94.5%

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

   8. *-commutative 94.5%

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

3. Simplified 94.5%

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

5. Taylor expanded in y around 0 53.1%

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

   1. *-commutative 53.1%

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

   2. *-commutative 53.1%

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

   3. associate-*r* 53.1%

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

   4. *-commutative 53.1%

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

7. Simplified 53.1%

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

   1. +-commutative 53.1%

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

   2. associate-*r* 53.1%

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

   3. distribute-rgt-out 56.2%

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

9. Applied egg-rr 56.2%

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

10. Final simplification 56.2%

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

Alternative 7: 15.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-+l+ 94.5%

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

   2. associate-*l* 94.5%

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

   3. *-commutative 94.5%

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

   4. *-commutative 94.5%

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

   5. +-commutative 94.5%

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

   6. fma-define 94.5%

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

   7. *-commutative 94.5%

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

   8. *-commutative 94.5%

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

3. Simplified 94.5%

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

5. Taylor expanded in y around 0 53.1%

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

   1. *-commutative 53.1%

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

   2. *-commutative 53.1%

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

   3. associate-*r* 53.1%

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

   4. *-commutative 53.1%

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

7. Simplified 53.1%

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

8. Taylor expanded in x around 0 11.1%

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

   1. *-commutative 11.1%

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

   2. associate-*r* 11.1%

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

   3. *-commutative 11.1%

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

10. Simplified 11.1%

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

11. Final simplification 11.1%

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

Developer target: 93.6% 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 2024046 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

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