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

Percentage Accurate: 99.9% → 100.0%

Time: 5.7s

Alternatives: 5

Speedup: 1.7×

• 43.0% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(y * 2.0), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * y), $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. distribute-lft-out N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. count-2 N/A

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

   9. distribute-rgt1-in N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

      \[\leadsto x \cdot x + \left(2 + 1\right) \cdot \left(\color{blue}{y} \cdot y\right) \]

   4. distribute-rgt1-in N/A

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

   5. count-2 N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-out N/A

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

   9. fma-define N/A

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

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

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

   11. count-2 N/A

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

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

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

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

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

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

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

   15. *-lowering-*.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.9%

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

Alternative 3: 81.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-89) (* x x) (* (* y y) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-89) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-89) then
        tmp = x * x
    else
        tmp = (y * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-89) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-89:
		tmp = x * x
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-89)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(y * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-89)
		tmp = x * x;
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-89], N[(x * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999967e-89

   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. distribute-lft-out N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. count-2 N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(y \cdot 3\right)} \]
   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 89.4%

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

   7. Simplified 89.4%

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

   if 4.99999999999999967e-89 < (*.f64 y y)

   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. distribute-lft-out N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. count-2 N/A

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

      9. distribute-rgt1-in N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot x + y \cdot \left(y \cdot 3\right)} \]
   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 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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. distribute-lft-out N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. count-2 N/A

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

   9. distribute-rgt1-in N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 5: 58.1% 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. distribute-lft-out N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. count-2 N/A

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

   9. distribute-rgt1-in N/A

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

   10. *-commutative N/A

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

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

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot x + y \cdot \left(y \cdot 3\right)} \]
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 60.2%

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

7. Simplified 60.2%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024150 
(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)))