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

Percentage Accurate: 99.9% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: 1.7×

• 43.2% 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+ 100.0%

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

   4. fma-def 100.0%

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

   5. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * x + 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+ 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. Final simplification 99.9%

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

Alternative 3: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-144} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.25e-144) (not (<= x 4.6e-84)))
   (+ (* y y) (* x x))
   (* (* y y) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.25e-144) || !(x <= 4.6e-84)) {
		tmp = (y * y) + (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 ((x <= (-1.25d-144)) .or. (.not. (x <= 4.6d-84))) then
        tmp = (y * y) + (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 ((x <= -1.25e-144) || !(x <= 4.6e-84)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.25e-144) or not (x <= 4.6e-84):
		tmp = (y * y) + (x * x)
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.25e-144) || !(x <= 4.6e-84))
		tmp = Float64(Float64(y * y) + 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 ((x <= -1.25e-144) || ~((x <= 4.6e-84)))
		tmp = (y * y) + (x * x);
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.25e-144], N[Not[LessEqual[x, 4.6e-84]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e-144 or 4.59999999999999961e-84 < x

   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. add-sqr-sqrt 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-def 99.9%

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

      6. hypot-def 99.9%

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

      7. +-commutative 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. hypot-def 99.9%

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

      10. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. unpow2 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 91.1%

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

      1. unpow2 91.1%

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

   8. Simplified 91.1%

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

   if -1.2499999999999999e-144 < x < 4.59999999999999961e-84

   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 96.0%

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

      1. unpow2 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*l* 96.0%

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

      4. *-commutative 96.0%

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

      5. count-2 96.0%

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

   4. Simplified 96.0%

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

      1. distribute-lft-out 95.9%

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

      2. count-2 95.9%

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

      3. metadata-eval 95.9%

         \[\leadsto y \cdot \left(\color{blue}{\sqrt{4}} \cdot y + y\right) \]

      4. *-un-lft-identity 95.9%

         \[\leadsto y \cdot \left(\sqrt{4} \cdot y + \color{blue}{1 \cdot y}\right) \]

      5. distribute-rgt-out 95.9%

         \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\sqrt{4} + 1\right)\right)} \]

      6. metadata-eval 95.9%

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

      7. metadata-eval 95.9%

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

      8. associate-*r* 96.0%

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

   6. Applied egg-rr 96.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-144} \lor \neg \left(x \leq 4.6 \cdot 10^{-84}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+101}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+101) (* x x) (* y (* y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+101) {
		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+101) 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+101) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+101:
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+101)
		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+101)
		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+101], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999989e101

   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 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

   if 4.99999999999999989e101 < (*.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. Taylor expanded in x around 0 88.9%

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

      1. unpow2 88.9%

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

      2. unpow2 88.9%

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

      3. distribute-rgt1-in 88.9%

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

      4. metadata-eval 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-*r* 88.9%

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

   4. Simplified 88.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+101}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 6: 75.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.7 \cdot 10^{+304}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.7e+304) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.7e+304) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	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) <= 1.7d+304) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.7e+304) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.7e+304:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.7e+304)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.7e+304)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.7e+304], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.7000000000000001e304

   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 inf 70.0%

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

      1. unpow2 70.0%

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

   4. Simplified 70.0%

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

   if 1.7000000000000001e304 < (*.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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-def 100.0%

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

      6. hypot-def 100.0%

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

      7. +-commutative 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. hypot-def 100.0%

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

      10. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. unpow2 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 97.9%

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

      1. unpow2 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in x around 0 97.9%

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

       1. unpow2 97.9%

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

   11. Simplified 97.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.7 \cdot 10^{+304}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 57.1% 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. Taylor expanded in x around inf 57.6%

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

   1. unpow2 57.6%

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

4. Simplified 57.6%

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

5. Final simplification 57.6%

   \[\leadsto x \cdot x \]

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 2023196 
(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)))