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

Percentage Accurate: 99.9% → 100.0%

Time: 3.8s

Alternatives: 6

Speedup: 1.7×

• 45.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * y + N[(x * x + N[(y * 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-def 99.9%

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

   3. associate-+l+ 99.9%

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

   4. fma-def 99.9%

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

   5. distribute-lft-out 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

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(y, Float64(y * 3.0), Float64(x * x))
end

Wolfram

code[x_, y_] := N[(y * N[(y * 3.0), $MachinePrecision] + 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. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. unpow2 99.9%

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

   3. associate-+r+ 99.9%

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

   4. unpow2 99.9%

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

   5. distribute-lft1-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-*r* 99.9%

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

   9. fma-def 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 4 \cdot 10^{+160}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+200}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-123)
   (* x x)
   (if (<= (* y y) 4e+160)
     (* 3.0 (* y y))
     (if (<= (* y y) 2e+200) (* x x) (* y (* y 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-123) {
		tmp = x * x;
	} else if ((y * y) <= 4e+160) {
		tmp = 3.0 * (y * y);
	} else if ((y * y) <= 2e+200) {
		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-123) then
        tmp = x * x
    else if ((y * y) <= 4d+160) then
        tmp = 3.0d0 * (y * y)
    else if ((y * y) <= 2d+200) 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-123) {
		tmp = x * x;
	} else if ((y * y) <= 4e+160) {
		tmp = 3.0 * (y * y);
	} else if ((y * y) <= 2e+200) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-123:
		tmp = x * x
	elif (y * y) <= 4e+160:
		tmp = 3.0 * (y * y)
	elif (y * y) <= 2e+200:
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-123)
		tmp = Float64(x * x);
	elseif (Float64(y * y) <= 4e+160)
		tmp = Float64(3.0 * Float64(y * y));
	elseif (Float64(y * y) <= 2e+200)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-123)
		tmp = x * x;
	elseif ((y * y) <= 4e+160)
		tmp = 3.0 * (y * y);
	elseif ((y * y) <= 2e+200)
		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-123], N[(x * x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 4e+160], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e+200], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 5.0000000000000003e-123 or 4.00000000000000003e160 < (*.f64 y y) < 1.9999999999999999e200

   1. Initial program 99.9%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around inf 84.1%

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

      1. unpow2 84.1%

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

   4. Simplified 84.1%

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

   if 5.0000000000000003e-123 < (*.f64 y y) < 4.00000000000000003e160

   1. Initial program 99.7%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around 0 65.7%

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

      1. unpow2 65.7%

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

      2. *-commutative 65.7%

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

      3. associate-*l* 65.7%

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

      4. *-commutative 65.7%

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

      5. count-2 65.7%

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

   4. Simplified 65.7%

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

      1. distribute-lft-out 65.7%

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

      2. count-2 65.7%

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

      3. *-un-lft-identity 65.7%

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

      4. distribute-rgt-out 65.7%

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

      5. metadata-eval 65.7%

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

      6. associate-*r* 65.7%

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

   6. Applied egg-rr 65.7%

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

   if 1.9999999999999999e200 < (*.f64 y y)

   1. Initial program 100.0%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around 0 94.2%

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

      1. unpow2 94.2%

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

      2. unpow2 94.2%

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

      3. distribute-rgt1-in 94.2%

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

      4. metadata-eval 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*r* 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-123}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 4 \cdot 10^{+160}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+200}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 4: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-62} \lor \neg \left(y \leq 2.05 \cdot 10^{+80}\right) \land y \leq 1.1 \cdot 10^{+100}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 7.5e-62) (and (not (<= y 2.05e+80)) (<= y 1.1e+100)))
   (* x x)
   (* y (* y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 7.5e-62) || (!(y <= 2.05e+80) && (y <= 1.1e+100))) {
		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 <= 7.5d-62) .or. (.not. (y <= 2.05d+80)) .and. (y <= 1.1d+100)) 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 <= 7.5e-62) || (!(y <= 2.05e+80) && (y <= 1.1e+100))) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 7.5e-62) or (not (y <= 2.05e+80) and (y <= 1.1e+100)):
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 7.5e-62) || (!(y <= 2.05e+80) && (y <= 1.1e+100)))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 7.5e-62) || (~((y <= 2.05e+80)) && (y <= 1.1e+100)))
		tmp = x * x;
	else
		tmp = y * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 7.5e-62], And[N[Not[LessEqual[y, 2.05e+80]], $MachinePrecision], LessEqual[y, 1.1e+100]]], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000003e-62 or 2.05000000000000001e80 < y < 1.1e100

   1. Initial program 99.9%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around inf 61.4%

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

      1. unpow2 61.4%

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

   4. Simplified 61.4%

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

   if 7.5000000000000003e-62 < y < 2.05000000000000001e80 or 1.1e100 < y

   1. Initial program 99.8%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around 0 86.1%

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

      1. unpow2 86.1%

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

      2. unpow2 86.1%

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

      3. distribute-rgt1-in 86.1%

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

      4. metadata-eval 86.1%

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

      5. *-commutative 86.1%

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

      6. associate-*r* 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-62} \lor \neg \left(y \leq 2.05 \cdot 10^{+80}\right) \land y \leq 1.1 \cdot 10^{+100}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 5: 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+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. count-2 99.9%

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

   5. distribute-rgt1-in 99.9%

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

   6. *-commutative 99.9%

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

   7. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-*l* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 6: 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. Taylor expanded in x around inf 53.2%

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

   1. unpow2 53.2%

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

4. Simplified 53.2%

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

5. Final simplification 53.2%

   \[\leadsto x \cdot x \]

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 2023214 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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