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

Percentage Accurate: 99.9% → 100.0%

Time: 7.3s

Alternatives: 5

Speedup: 1.2×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * y + N[(x * x + N[(2.0 * 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%

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

   2. sqr-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. sqr-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

   7. sqr-neg 99.9%

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

   8. sqr-neg 99.9%

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

   9. associate-+l+ 99.9%

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

   10. fma-define 100.0%

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

   11. count-2 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

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(y * Float64(y * 3.0)))
end

Wolfram

code[x_, y_] := N[(x * x + 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-define 99.9%

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

   4. *-lft-identity 99.9%

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

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   9. 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. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.7%

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

   2. sqrt-unprod 87.1%

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

   3. swap-sqr 87.1%

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

   4. pow2 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{\left(\color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right) \cdot \left(3 \cdot 3\right)}\right) \]

   5. pow2 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{\left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \cdot \left(3 \cdot 3\right)}\right) \]

   6. pow-prod-up 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{\color{blue}{{y}^{\left(2 + 2\right)}} \cdot \left(3 \cdot 3\right)}\right) \]

   7. metadata-eval 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{{y}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)}\right) \]

   8. metadata-eval 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{{y}^{4} \cdot \color{blue}{9}}\right) \]

6. Applied egg-rr 87.1%

   \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\sqrt{{y}^{4} \cdot 9}}\right) \]
7. Step-by-step derivation

   1. sqrt-prod 87.1%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\sqrt{{y}^{4}} \cdot \sqrt{9}}\right) \]

   2. sqrt-pow1 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{9}\right) \]

   3. metadata-eval 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, {y}^{\color{blue}{2}} \cdot \sqrt{9}\right) \]

   4. pow2 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot y\right)} \cdot \sqrt{9}\right) \]

   5. associate-*l* 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot \sqrt{9}\right)}\right) \]

   6. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 3: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{-72} \lor \neg \left(x \cdot x \leq 1.95 \cdot 10^{+61}\right) \land x \cdot x \leq 1.7 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 6.5e-72)
         (and (not (<= (* x x) 1.95e+61)) (<= (* x x) 1.7e+105)))
   (* y (* y 3.0))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 6.5e-72) || (!((x * x) <= 1.95e+61) && ((x * x) <= 1.7e+105))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * x) <= 6.5d-72) .or. (.not. ((x * x) <= 1.95d+61)) .and. ((x * x) <= 1.7d+105)) then
        tmp = y * (y * 3.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) <= 6.5e-72) || (!((x * x) <= 1.95e+61) && ((x * x) <= 1.7e+105))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 6.5e-72) or (not ((x * x) <= 1.95e+61) and ((x * x) <= 1.7e+105)):
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 6.5e-72) || (!(Float64(x * x) <= 1.95e+61) && (Float64(x * x) <= 1.7e+105)))
		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) <= 6.5e-72) || (~(((x * x) <= 1.95e+61)) && ((x * x) <= 1.7e+105)))
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 6.5e-72], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.95e+61]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1.7e+105]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 6.4999999999999997e-72 or 1.94999999999999994e61 < (*.f64 x x) < 1.7e105

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. fma-define 99.8%

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

      4. *-lft-identity 99.8%

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

      5. metadata-eval 99.8%

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

      6. count-2 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.5%

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

      2. fma-undefine 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. hypot-define 99.5%

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

      5. sqrt-prod 99.6%

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

      6. sqrt-prod 53.0%

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

      7. add-sqr-sqrt 99.6%

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

      8. fma-undefine 99.6%

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

      9. add-sqr-sqrt 99.6%

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

      10. hypot-define 99.6%

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

      11. sqrt-prod 99.4%

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

      12. sqrt-prod 52.9%

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

      13. add-sqr-sqrt 99.4%

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

   6. Applied egg-rr 99.4%

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

   8. Simplified 99.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot \sqrt{3}\right)\right)}^{2}} \]

   9. Taylor expanded in x around 0 85.8%

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

       1. unpow-prod-down 85.7%

          \[\leadsto \color{blue}{{y}^{2} \cdot {\left(\sqrt{3}\right)}^{2}} \]

       2. pow2 85.7%

          \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot {\left(\sqrt{3}\right)}^{2} \]

       3. pow2 85.7%

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

       4. rem-square-sqrt 86.2%

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

       5. associate-*l* 86.2%

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

   11. Applied egg-rr 86.2%

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

   if 6.4999999999999997e-72 < (*.f64 x x) < 1.94999999999999994e61 or 1.7e105 < (*.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 85.7%

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

      1. pow2 99.9%

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

   5. Applied egg-rr 85.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6.5 \cdot 10^{-72} \lor \neg \left(x \cdot x \leq 1.95 \cdot 10^{+61}\right) \land x \cdot x \leq 1.7 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $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. count-2 99.9%

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

   4. add-sqr-sqrt 99.8%

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

   5. fma-define 99.8%

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

   6. *-commutative 99.8%

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

   7. sqrt-prod 99.8%

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

   8. sqrt-prod 51.5%

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

   9. add-sqr-sqrt 75.7%

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

   10. *-commutative 75.7%

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

   11. sqrt-prod 75.7%

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

   12. sqrt-prod 51.4%

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

   13. add-sqr-sqrt 99.7%

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

   14. pow2 99.7%

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

4. Applied egg-rr 99.7%

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

6. Simplified 99.7%

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

   1. pow2 99.7%

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

8. Applied egg-rr 99.7%

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

   1. unpow-prod-down 99.6%

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

   2. pow2 99.6%

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

   3. pow2 99.6%

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

   4. rem-square-sqrt 99.9%

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

   5. associate-*l* 99.9%

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

   6. *-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 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 58.8%

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

   1. pow2 99.7%

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

5. Applied egg-rr 58.8%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024110 
(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)))