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

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 7

Speedup: 1.7×

• 43.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(y + y), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-out 99.9%

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

   4. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * 2.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. *-un-lft-identity 99.9%

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

   5. distribute-rgt-out 99.9%

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

   6. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 7e-50) (* (* y y) 3.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 7e-50) {
		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) <= 7d-50) 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) <= 7e-50) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 7e-50:
		tmp = (y * y) * 3.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 7e-50)
		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) <= 7e-50)
		tmp = (y * y) * 3.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 7e-50], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.99999999999999993e-50

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. unpow2 82.3%

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

   5. Simplified 82.3%

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

      1. *-un-lft-identity 82.3%

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

      2. distribute-rgt-out 82.3%

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

      3. metadata-eval 82.3%

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

   7. Applied egg-rr 82.3%

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

   if 6.99999999999999993e-50 < (*.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. Add Preprocessing

   3. Taylor expanded in x around inf 84.7%

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

      1. unpow2 84.7%

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

   5. Simplified 84.7%

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

Alternative 4: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.9 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2.9e-50) (* y (* y 3.0)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.9e-50) {
		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) <= 2.9d-50) 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) <= 2.9e-50) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2.9e-50:
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2.9e-50)
		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) <= 2.9e-50)
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.9e-50], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.90000000000000008e-50

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. unpow2 82.3%

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

      2. unpow2 82.3%

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

      3. distribute-lft1-in 82.3%

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

      4. metadata-eval 82.3%

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

      5. *-commutative 82.3%

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

      6. associate-*r* 82.3%

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

   5. Simplified 82.3%

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

   if 2.90000000000000008e-50 < (*.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. Add Preprocessing

   3. Taylor expanded in x around inf 84.7%

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

      1. unpow2 84.7%

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

   5. Simplified 84.7%

      \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-out 99.9%

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

   4. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. count-2 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-define 99.9%

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

   4. associate-*l* 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. *-commutative 99.9%

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

   9. *-un-lft-identity 99.9%

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

   10. distribute-rgt-out 99.9%

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

   11. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-out 99.9%

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

   4. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. count-2 100.0%

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

   2. *-commutative 100.0%

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

   3. fma-define 99.9%

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

   4. associate-*l* 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. associate-+r+ 99.9%

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

   8. *-commutative 99.9%

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

   9. *-un-lft-identity 99.9%

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

   10. distribute-rgt-out 99.9%

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

   11. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-*l* 99.9%

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

   2. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 7: 57.8% 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. Add Preprocessing

3. Taylor expanded in x around inf 61.3%

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

   1. unpow2 61.3%

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

5. Simplified 61.3%

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

Developer target: 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 2024097 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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