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

Percentage Accurate: 99.9% → 100.0%

Time: 4.7s

Alternatives: 5

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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-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. count-2 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. count-2 99.9%

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

   7. associate-*r* 99.9%

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

   8. *-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 88.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{-307} \lor \neg \left(x \cdot x \leq 5.6 \cdot 10^{-129}\right) \land x \cdot x \leq 1.35 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 6e-307)
         (and (not (<= (* x x) 5.6e-129)) (<= (* x x) 1.35e-23)))
   (* y (* y 3.0))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 6e-307) || (!((x * x) <= 5.6e-129) && ((x * x) <= 1.35e-23))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) + (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 (((x * x) <= 6d-307) .or. (.not. ((x * x) <= 5.6d-129)) .and. ((x * x) <= 1.35d-23)) then
        tmp = y * (y * 3.0d0)
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) <= 6e-307) || (!((x * x) <= 5.6e-129) && ((x * x) <= 1.35e-23))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 6e-307) or (not ((x * x) <= 5.6e-129) and ((x * x) <= 1.35e-23)):
		tmp = y * (y * 3.0)
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 6e-307) || (!(Float64(x * x) <= 5.6e-129) && (Float64(x * x) <= 1.35e-23)))
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * x) <= 6e-307) || (~(((x * x) <= 5.6e-129)) && ((x * x) <= 1.35e-23)))
		tmp = y * (y * 3.0);
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 6e-307], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 5.6e-129]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1.35e-23]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.9999999999999999e-307 or 5.5999999999999998e-129 < (*.f64 x x) < 1.34999999999999992e-23

   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 94.3%

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

   4. Simplified 94.4%

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

   if 5.9999999999999999e-307 < (*.f64 x x) < 5.5999999999999998e-129 or 1.34999999999999992e-23 < (*.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. Step-by-step derivation

      1. associate-+l+ 100.0%

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

      2. count-2 100.0%

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

      3. flip-+ 18.5%

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

      4. pow2 18.5%

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

      5. pow2 18.5%

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

      6. pow-prod-up 18.4%

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

      7. metadata-eval 18.4%

         \[\leadsto \frac{{x}^{\color{blue}{4}} - \left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      8. *-commutative 18.4%

         \[\leadsto \frac{{x}^{4} - \color{blue}{\left(\left(y \cdot y\right) \cdot 2\right)} \cdot \left(2 \cdot \left(y \cdot y\right)\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      9. *-commutative 18.4%

         \[\leadsto \frac{{x}^{4} - \left(\left(y \cdot y\right) \cdot 2\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot 2\right)}}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      10. swap-sqr 18.4%

          \[\leadsto \frac{{x}^{4} - \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot 2\right)}}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      11. pow2 18.4%

          \[\leadsto \frac{{x}^{4} - \left(\color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot 2\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      12. pow2 18.4%

          \[\leadsto \frac{{x}^{4} - \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \cdot \left(2 \cdot 2\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      13. pow-prod-up 18.4%

          \[\leadsto \frac{{x}^{4} - \color{blue}{{y}^{\left(2 + 2\right)}} \cdot \left(2 \cdot 2\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      14. metadata-eval 18.4%

          \[\leadsto \frac{{x}^{4} - {y}^{\color{blue}{4}} \cdot \left(2 \cdot 2\right)}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      15. metadata-eval 18.4%

          \[\leadsto \frac{{x}^{4} - {y}^{4} \cdot \color{blue}{4}}{x \cdot x - 2 \cdot \left(y \cdot y\right)} + y \cdot y \]

      16. *-commutative 18.4%

          \[\leadsto \frac{{x}^{4} - {y}^{4} \cdot 4}{x \cdot x - \color{blue}{\left(y \cdot y\right) \cdot 2}} + y \cdot y \]

      17. associate-*l* 18.4%

          \[\leadsto \frac{{x}^{4} - {y}^{4} \cdot 4}{x \cdot x - \color{blue}{y \cdot \left(y \cdot 2\right)}} + y \cdot y \]

   3. Applied egg-rr 18.4%

      \[\leadsto \color{blue}{\frac{{x}^{4} - {y}^{4} \cdot 4}{x \cdot x - y \cdot \left(y \cdot 2\right)}} + y \cdot y \]

   4. Taylor expanded in x around inf 90.3%

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

      1. unpow2 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 6 \cdot 10^{-307} \lor \neg \left(x \cdot x \leq 5.6 \cdot 10^{-129}\right) \land x \cdot x \leq 1.35 \cdot 10^{-23}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 3: 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. 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. count-2 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. count-2 99.9%

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

   7. associate-*r* 99.9%

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

   8. *-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

   2. associate-*r* 99.9%

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

   3. *-commutative 99.9%

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

   4. *-un-lft-identity 99.9%

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

   5. distribute-rgt-out 99.9%

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

   6. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 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. +-commutative 99.9%

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

   4. associate-+r+ 99.9%

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

   5. count-2 99.9%

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

   6. distribute-lft1-in 99.9%

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

   7. metadata-eval 99.9%

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

   8. *-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. 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 5: 56.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * N[(y * 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. Taylor expanded in x around 0 56.3%

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

4. Simplified 56.4%

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

5. Final simplification 56.4%

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

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 2023268 
(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)))