Linear.Quaternion:$c/ from linear-1.19.1.3, E

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

Alternatives: 7

Speedup: 1.7×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot y, 3, x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* y y) 3.0 (* x x)))

C

double code(double x, double y) {
	return fma((y * y), 3.0, (x * x));
}

Julia

function code(x, y)
	return fma(Float64(y * y), 3.0, Float64(x * x))
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * 3.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot 3 + \color{blue}{x \cdot x} \]

   2. fma-define N/A

      \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{3}, x \cdot x\right) \]

   3. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(y \cdot y\right), \color{blue}{3}, \left(x \cdot x\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, y\right), 3, \left(x \cdot x\right)\right) \]

   5. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(y, y\right), 3, \mathsf{*.f64}\left(x, x\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, 3, x \cdot x\right)} \]
7. Add Preprocessing

Alternative 2: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{y \cdot -9}{\frac{-3}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2.8e+22) (/ (* y -9.0) (/ -3.0 y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.8e+22) {
		tmp = (y * -9.0) / (-3.0 / 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 * x) <= 2.8d+22) then
        tmp = (y * (-9.0d0)) / ((-3.0d0) / y)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.8e+22) {
		tmp = (y * -9.0) / (-3.0 / y);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2.8e+22:
		tmp = (y * -9.0) / (-3.0 / y)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2.8e+22)
		tmp = Float64(Float64(y * -9.0) / Float64(-3.0 / y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2.8e+22)
		tmp = (y * -9.0) / (-3.0 / y);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.8e+22], N[(N[(y * -9.0), $MachinePrecision] / N[(-3.0 / y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.8e22

   1. Initial program 99.7%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(\left(y \cdot y\right) \cdot 3\right) \cdot \left(\left(y \cdot y\right) \cdot 3\right)}{\color{blue}{x \cdot x - \left(y \cdot y\right) \cdot 3}} \]

      2. fmm-def N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(\left(y \cdot y\right) \cdot 3\right) \cdot \left(\left(y \cdot y\right) \cdot 3\right)}{\mathsf{fma}\left(x, \color{blue}{x}, \mathsf{neg}\left(\left(y \cdot y\right) \cdot 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(\left(y \cdot y\right) \cdot 3\right) \cdot \left(\left(y \cdot y\right) \cdot 3\right)}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(3 \cdot \left(y \cdot y\right)\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(\left(y \cdot y\right) \cdot 3\right) \cdot \left(\left(y \cdot y\right) \cdot 3\right)\right), \color{blue}{\left(\mathsf{fma}\left(x, x, \mathsf{neg}\left(3 \cdot \left(y \cdot y\right)\right)\right)\right)}\right) \]

   6. Applied egg-rr 39.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -9}{x \cdot x + \left(y \cdot y\right) \cdot -3}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \color{blue}{\left(-3 \cdot {y}^{2}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \left(-3 \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \left(\left(-3 \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \left(y \cdot \color{blue}{\left(-3 \cdot y\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(-3 \cdot y\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{-3}\right)\right)\right) \]

      6. *-lowering-*.f64 33.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), -9\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{-3}\right)\right)\right) \]

   9. Simplified 33.8%

      \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot -9}{\color{blue}{y \cdot \left(y \cdot -3\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-9 \cdot {y}^{4}\right)}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \left({y}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \left({y}^{\left(2 \cdot 2\right)}\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \left({y}^{2} \cdot {y}^{2}\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

       8. *-lowering-*.f64 33.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, -3\right)\right)\right) \]

   12. Simplified 33.8%

       \[\leadsto \frac{\color{blue}{-9 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}}{y \cdot \left(y \cdot -3\right)} \]
   13. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \left(y \cdot -3\right)}{-9 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{\left(y \cdot y\right) \cdot -3}{\color{blue}{-9} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{\left(y \cdot y\right) \cdot -3}{-9 \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \color{blue}{y}\right)}} \]

       4. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{\left(y \cdot y\right) \cdot -3}{-9 \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{\left(y \cdot y\right) \cdot -3}{\left(-9 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \color{blue}{y}}} \]

       6. times-frac N/A

          \[\leadsto \frac{1}{\frac{y \cdot y}{-9 \cdot \left(y \cdot \left(y \cdot y\right)\right)} \cdot \color{blue}{\frac{-3}{y}}} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{\frac{y \cdot y}{-9 \cdot \left(y \cdot \left(y \cdot y\right)\right)}}}{\color{blue}{\frac{-3}{y}}} \]

       8. clear-num N/A

          \[\leadsto \frac{\frac{-9 \cdot \left(y \cdot \left(y \cdot y\right)\right)}{y \cdot y}}{\frac{\color{blue}{-3}}{y}} \]

       9. associate-/l* N/A

          \[\leadsto \frac{-9 \cdot \frac{y \cdot \left(y \cdot y\right)}{y \cdot y}}{\frac{\color{blue}{-3}}{y}} \]

       10. cube-unmult N/A

           \[\leadsto \frac{-9 \cdot \frac{{y}^{3}}{y \cdot y}}{\frac{-3}{y}} \]

       11. pow2 N/A

           \[\leadsto \frac{-9 \cdot \frac{{y}^{3}}{{y}^{2}}}{\frac{-3}{y}} \]

       12. pow-div N/A

           \[\leadsto \frac{-9 \cdot {y}^{\left(3 - 2\right)}}{\frac{-3}{y}} \]

       13. metadata-eval N/A

           \[\leadsto \frac{-9 \cdot {y}^{1}}{\frac{-3}{y}} \]

       14. unpow1 N/A

           \[\leadsto \frac{-9 \cdot y}{\frac{-3}{y}} \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(-9 \cdot y\right), \color{blue}{\left(\frac{-3}{y}\right)}\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, y\right), \left(\frac{\color{blue}{-3}}{y}\right)\right) \]

       17. /-lowering-/.f64 83.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-9, y\right), \mathsf{/.f64}\left(-3, \color{blue}{y}\right)\right) \]

   14. Applied egg-rr 83.4%

       \[\leadsto \color{blue}{\frac{-9 \cdot y}{\frac{-3}{y}}} \]

   if 2.8e22 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 84.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{y \cdot -9}{\frac{-3}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5.5e+21) (* y (* y 3.0)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5.5e+21) {
		tmp = y * (y * 3.0);
	} 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 * x) <= 5.5d+21) then
        tmp = y * (y * 3.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5.5e+21) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5.5e+21:
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5.5e+21)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5.5e+21)
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.5e+21], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.5e21

   1. Initial program 99.7%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 83.3%

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

      1. associate-*r* N/A

         \[\leadsto \left(3 \cdot y\right) \cdot \color{blue}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot 3\right) \cdot y \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot 3\right), \color{blue}{y}\right) \]

      4. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 3\right), y\right) \]

   9. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left(y \cdot 3\right) \cdot y} \]

   if 5.5e21 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 84.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{+21}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 6.5e+21) (* (* y y) 3.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 6.5e+21) {
		tmp = (y * y) * 3.0;
	} 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 * x) <= 6.5d+21) then
        tmp = (y * y) * 3.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 6.5e+21) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 6.5e+21:
		tmp = (y * y) * 3.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 6.5e+21)
		tmp = Float64(Float64(y * y) * 3.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 6.5e+21)
		tmp = (y * y) * 3.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 6.5e+21], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.5e21

   1. Initial program 99.7%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto \color{blue}{3 \cdot \left(y \cdot y\right)} \]

   if 6.5e21 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 84.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{+21}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y\right) \cdot 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (* (* y y) 3.0)))

C

double code(double x, double y) {
	return (x * x) + ((y * y) * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * x) + ((y * y) * 3.0);
}

Python

def code(x, y):
	return (x * x) + ((y * y) * 3.0)

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(Float64(y * y) * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot \left(y \cdot 3\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x x) (* y (* y 3.0))))

C

double code(double x, double y) {
	return (x * x) + (y * (y * 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + (y * (y * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * x) + (y * (y * 3.0));
}

Python

def code(x, y):
	return (x * x) + (y * (y * 3.0))

Julia

function code(x, y)
	return Float64(Float64(x * x) + Float64(y * Float64(y * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + (y * (y * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(y \cdot 3\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot 3\right) \cdot \color{blue}{y}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot 3\right), \color{blue}{y}\right)\right) \]

   4. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, 3\right), y\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto x \cdot x + \color{blue}{\left(y \cdot 3\right) \cdot y} \]

7. Final simplification 99.9%

   \[\leadsto x \cdot x + y \cdot \left(y \cdot 3\right) \]
8. Add Preprocessing

Alternative 7: 57.7% accurate, 5.0× 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 99.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(y \cdot y + \left(y \cdot y + y \cdot y\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{y \cdot y} + \left(y \cdot y + y \cdot y\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y + 2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(y \cdot y\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), 3\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot x + \left(y \cdot y\right) \cdot 3} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 55.4%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

7. Simplified 55.4%

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

Developer Target 1: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (* y (+ y (+ y y)))))

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