Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 36.3s

Alternatives: 13

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.exp((y * (x * y)));
}

Python

def code(x, y):
	return math.exp((y * (x * y)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto e^{y \cdot \left(x \cdot y\right)} \]
4. Add Preprocessing

Alternative 2: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{-72}:\\ \;\;\;\;1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+107}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.35e-72)
   (+
    1.0
    (*
     (* y (* x y))
     (+ 1.0 (* y (* y (* (* x (* x (* y y))) 0.16666666666666666))))))
   (if (<= y 8.6e+107) (exp x) (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-72) {
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	} else if (y <= 8.6e+107) {
		tmp = exp(x);
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 <= 3.35d-72) then
        tmp = 1.0d0 + ((y * (x * y)) * (1.0d0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666d0)))))
    else if (y <= 8.6d+107) then
        tmp = exp(x)
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-72) {
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	} else if (y <= 8.6e+107) {
		tmp = Math.exp(x);
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.35e-72:
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))))
	elif y <= 8.6e+107:
		tmp = math.exp(x)
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.35e-72)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x * y)) * Float64(1.0 + Float64(y * Float64(y * Float64(Float64(x * Float64(x * Float64(y * y))) * 0.16666666666666666))))));
	elseif (y <= 8.6e+107)
		tmp = exp(x);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.35e-72)
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	elseif (y <= 8.6e+107)
		tmp = exp(x);
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.35e-72], N[(1.0 + N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+107], N[Exp[x], $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.35e-72

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)} \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{\color{blue}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right), \color{blue}{\left(\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 76.0%

      \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \cdot 0.027777777777777776\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{0.5 - \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666}}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left({y}^{3} \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({x}^{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. Simplified 83.4%

       \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)}\right) \]

   if 3.35e-72 < y < 8.5999999999999999e107

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 88.5%

      \[\leadsto e^{\color{blue}{x}} \]

   if 8.5999999999999999e107 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + \color{blue}{1} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + x \cdot {y}^{2}\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + \color{blue}{\left(x \cdot {y}^{2} + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}\right) + \left(x \cdot \color{blue}{{y}^{2}} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(\color{blue}{x \cdot {y}^{2}} + 1\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(1 + \color{blue}{x \cdot {y}^{2}}\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot {y}^{4}\right), \color{blue}{\frac{1}{2}}, 1 + x \cdot {y}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 42.3%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.5\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({y}^{4} \cdot \color{blue}{{x}^{2}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot \color{blue}{x} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-83}:\\ \;\;\;\;1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-83)
   (+
    1.0
    (*
     (* y (* x y))
     (+ 1.0 (* y (* y (* (* x (* x (* y y))) 0.16666666666666666))))))
   (exp (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-83) {
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	} else {
		tmp = exp((x * 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 <= 3.1d-83) then
        tmp = 1.0d0 + ((y * (x * y)) * (1.0d0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666d0)))))
    else
        tmp = exp((x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-83) {
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	} else {
		tmp = Math.exp((x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-83:
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))))
	else:
		tmp = math.exp((x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-83)
		tmp = Float64(1.0 + Float64(Float64(y * Float64(x * y)) * Float64(1.0 + Float64(y * Float64(y * Float64(Float64(x * Float64(x * Float64(y * y))) * 0.16666666666666666))))));
	else
		tmp = exp(Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-83)
		tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
	else
		tmp = exp((x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-83], N[(1.0 + N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999992e-83

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Simplified 82.7%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)} \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{\color{blue}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}}\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right), \color{blue}{\left(\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 75.8%

      \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \cdot 0.027777777777777776\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{0.5 - \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666}}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left({y}^{3} \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({x}^{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. Simplified 83.3%

       \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)}\right) \]

   if 3.09999999999999992e-83 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 85.1%

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

Alternative 4: 68.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-84}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+107}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-84)
   (+ 1.0 (* y (* x y)))
   (if (<= y 7e+107)
     (+ 1.0 (* x (* x (* (* y y) (+ 0.5 (* x (* y 0.16666666666666666)))))))
     (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-84) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 7e+107) {
		tmp = 1.0 + (x * (x * ((y * y) * (0.5 + (x * (y * 0.16666666666666666))))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 <= 1.15d-84) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 7d+107) then
        tmp = 1.0d0 + (x * (x * ((y * y) * (0.5d0 + (x * (y * 0.16666666666666666d0))))))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-84) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 7e+107) {
		tmp = 1.0 + (x * (x * ((y * y) * (0.5 + (x * (y * 0.16666666666666666))))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.15e-84:
		tmp = 1.0 + (y * (x * y))
	elif y <= 7e+107:
		tmp = 1.0 + (x * (x * ((y * y) * (0.5 + (x * (y * 0.16666666666666666))))))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-84)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 7e+107)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(Float64(y * y) * Float64(0.5 + Float64(x * Float64(y * 0.16666666666666666)))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-84)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 7e+107)
		tmp = 1.0 + (x * (x * ((y * y) * (0.5 + (x * (y * 0.16666666666666666))))));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.15e-84], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+107], N[(1.0 + N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * N[(0.5 + N[(x * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1499999999999999e-84

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 75.8%

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

   if 1.1499999999999999e-84 < y < 6.9999999999999995e107

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 95.6%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{{y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left({y}^{3} \cdot \frac{1}{6}\right) \cdot {x}^{2} + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot {x}^{2} + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \left(x \cdot x\right) + {y}^{\color{blue}{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot x\right) \cdot x + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left({y}^{3} \cdot x\right)\right) \cdot x + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + \left({y}^{3} \cdot \frac{x}{y}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   10. Simplified 64.4%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 \cdot y\right)\right)\right)\right)} \]

   if 6.9999999999999995e107 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + \color{blue}{1} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + x \cdot {y}^{2}\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + \color{blue}{\left(x \cdot {y}^{2} + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}\right) + \left(x \cdot \color{blue}{{y}^{2}} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(\color{blue}{x \cdot {y}^{2}} + 1\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(1 + \color{blue}{x \cdot {y}^{2}}\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot {y}^{4}\right), \color{blue}{\frac{1}{2}}, 1 + x \cdot {y}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 42.3%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.5\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({y}^{4} \cdot \color{blue}{{x}^{2}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot \color{blue}{x} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-84}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+107}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-84}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+106}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e-84)
   (+ 1.0 (* y (* x y)))
   (if (<= y 5.2e+106)
     (+ 1.0 (* (* y y) (* 0.5 (* x x))))
     (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-84) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 5.2e+106) {
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 <= 6.4d-84) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 5.2d+106) then
        tmp = 1.0d0 + ((y * y) * (0.5d0 * (x * x)))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-84) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 5.2e+106) {
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.4e-84:
		tmp = 1.0 + (y * (x * y))
	elif y <= 5.2e+106:
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.4e-84)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 5.2e+106)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(0.5 * Float64(x * x))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e-84)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 5.2e+106)
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.4e-84], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+106], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.3999999999999999e-84

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 75.8%

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

   if 6.3999999999999999e-84 < y < 5.20000000000000039e106

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 95.6%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{{y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left({y}^{3} \cdot \frac{1}{6}\right) \cdot {x}^{2} + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot {x}^{2} + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \left(x \cdot x\right) + {y}^{\color{blue}{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot x\right) \cdot x + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left({y}^{3} \cdot x\right)\right) \cdot x + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + \left({y}^{3} \cdot \frac{x}{y}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   10. Simplified 64.4%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 \cdot y\right)\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{2}} \cdot {x}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{2}} \cdot {x}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right) \]

       10. *-lowering-*.f64 58.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right) \]

   13. Simplified 58.4%

       \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)} \]

   if 5.20000000000000039e106 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + \color{blue}{1} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + x \cdot {y}^{2}\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + \color{blue}{\left(x \cdot {y}^{2} + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}\right) + \left(x \cdot \color{blue}{{y}^{2}} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(\color{blue}{x \cdot {y}^{2}} + 1\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(1 + \color{blue}{x \cdot {y}^{2}}\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot {y}^{4}\right), \color{blue}{\frac{1}{2}}, 1 + x \cdot {y}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 42.3%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.5\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({y}^{4} \cdot \color{blue}{{x}^{2}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot \color{blue}{x} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-84}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+106}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (*
   (* y (* x y))
   (+ 1.0 (* y (* y (* (* x (* x (* y y))) 0.16666666666666666)))))))

C

double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * (x * y)) * (1.0d0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666d0)))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
}

Python

def code(x, y):
	return 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * Float64(x * y)) * Float64(1.0 + Float64(y * Float64(y * Float64(Float64(x * Float64(x * Float64(y * y))) * 0.16666666666666666))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * (x * y)) * (1.0 + (y * (y * ((x * (x * (y * y))) * 0.16666666666666666)))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)} \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{\color{blue}{\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right), \color{blue}{\left(\frac{1}{2} - x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

7. Applied egg-rr 64.6%

   \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \cdot 0.027777777777777776\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{0.5 - \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666}}\right) \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   13. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \left({y}^{3} \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   15. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\left(\frac{1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({x}^{2} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

10. Simplified 75.9%

    \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot \left(\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.16666666666666666\right)\right)}\right) \]
11. Add Preprocessing

Alternative 7: 71.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (*
   (* y (* x y))
   (+ 1.0 (* x (* x (* 0.16666666666666666 (* y (* y (* y y))))))))))

C

double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * (x * y)) * (1.0d0 + (x * (x * (0.16666666666666666d0 * (y * (y * (y * y))))))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
}

Python

def code(x, y):
	return 1.0 + ((y * (x * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * Float64(x * y)) * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(y * Float64(y * Float64(y * y)))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * (x * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left({x}^{2} \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(x \cdot x\right) \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {y}^{4}\right)\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({y}^{4} \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   13. pow-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   19. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   22. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   23. *-lowering-*.f64 74.8%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 74.8%

   \[\leadsto 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)}\right) \]
9. Add Preprocessing

Alternative 8: 70.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq 7.8 \cdot 10^{+107}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + t\_0 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= y 7.8e+107)
     (+ 1.0 (* t_0 (+ 1.0 (* t_0 0.5))))
     (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (y <= 7.8e+107) {
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (y <= 7.8d+107) then
        tmp = 1.0d0 + (t_0 * (1.0d0 + (t_0 * 0.5d0)))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (y <= 7.8e+107) {
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if y <= 7.8e+107:
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (y <= 7.8e+107)
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(t_0 * 0.5))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (y <= 7.8e+107)
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.8e+107], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.7999999999999997e107

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + \color{blue}{1} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + x \cdot {y}^{2}\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + \color{blue}{\left(x \cdot {y}^{2} + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}\right) + \left(x \cdot \color{blue}{{y}^{2}} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(\color{blue}{x \cdot {y}^{2}} + 1\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(1 + \color{blue}{x \cdot {y}^{2}}\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot {y}^{4}\right), \color{blue}{\frac{1}{2}}, 1 + x \cdot {y}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 76.6%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.5\right)} \]

   if 7.7999999999999997e107 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + \color{blue}{1} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + x \cdot {y}^{2}\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right) + \color{blue}{\left(x \cdot {y}^{2} + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{2}\right) + \left(x \cdot \color{blue}{{y}^{2}} + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(\color{blue}{x \cdot {y}^{2}} + 1\right) \]

      6. +-commutative N/A

         \[\leadsto \left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{2} + \left(1 + \color{blue}{x \cdot {y}^{2}}\right) \]

      7. fma-define N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \left(x \cdot {y}^{4}\right), \color{blue}{\frac{1}{2}}, 1 + x \cdot {y}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 42.3%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.5\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({y}^{4} \cdot \color{blue}{{x}^{2}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot \color{blue}{x} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-83}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+134}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-83)
   (+ 1.0 (* y (* x y)))
   (if (<= y 3.7e+134)
     (+ 1.0 (* (* y y) (* 0.5 (* x x))))
     (* (* x (* x (* y y))) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-83) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.7e+134) {
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	} else {
		tmp = (x * (x * (y * y))) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-83) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 3.7d+134) then
        tmp = 1.0d0 + ((y * y) * (0.5d0 * (x * x)))
    else
        tmp = (x * (x * (y * y))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-83) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.7e+134) {
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	} else {
		tmp = (x * (x * (y * y))) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-83:
		tmp = 1.0 + (y * (x * y))
	elif y <= 3.7e+134:
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)))
	else:
		tmp = (x * (x * (y * y))) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-83)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 3.7e+134)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(0.5 * Float64(x * x))));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(y * y))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-83)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 3.7e+134)
		tmp = 1.0 + ((y * y) * (0.5 * (x * x)));
	else
		tmp = (x * (x * (y * y))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-83], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+134], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.2000000000000001e-83

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 75.8%

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

   if 3.2000000000000001e-83 < y < 3.70000000000000013e134

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 89.8%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]

   7. Simplified 61.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(y + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left({y}^{3} \cdot \left(\frac{1}{6} \cdot {x}^{2}\right) + \color{blue}{{y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left({y}^{3} \cdot \frac{1}{6}\right) \cdot {x}^{2} + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot {x}^{2} + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \left(x \cdot x\right) + {y}^{\color{blue}{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{6} \cdot {y}^{3}\right) \cdot x\right) \cdot x + \color{blue}{{y}^{3}} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left({y}^{3} \cdot x\right)\right) \cdot x + {\color{blue}{y}}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + {y}^{3} \cdot \left(\frac{x}{y} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right) \cdot x + \left({y}^{3} \cdot \frac{x}{y}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   10. Simplified 61.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(y \cdot y\right) \cdot \left(0.5 + x \cdot \left(0.16666666666666666 \cdot y\right)\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{2}} \cdot {x}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{2}} \cdot {x}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right)\right)\right) \]

       10. *-lowering-*.f64 54.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right)\right)\right) \]

   13. Simplified 54.7%

       \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)} \]

   if 3.70000000000000013e134 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 76.6%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(x + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)\right)}\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right) \cdot y}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)} \cdot y\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \frac{1}{2} \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot y\right)}\right)\right) \]

   7. Simplified 32.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(y \cdot 0.5\right) + 1\right) \cdot y\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot {\color{blue}{y}}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {y}^{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 49.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 49.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-83}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+134}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+145}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e+145) (+ 1.0 (* y (* x y))) (* (* x (* x (* y y))) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e+145) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = (x * (x * (y * y))) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d+145) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = (x * (x * (y * y))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e+145) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = (x * (x * (y * y))) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e+145:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = (x * (x * (y * y))) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e+145)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(y * y))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e+145)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = (x * (x * (y * y))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e+145], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999999e145

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 68.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 68.5%

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

   if 7.9999999999999999e145 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 77.6%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(x + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)\right)}\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right) \cdot y}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)} \cdot y\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \frac{1}{2} \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot y\right)}\right)\right) \]

   7. Simplified 36.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(y \cdot 0.5\right) + 1\right) \cdot y\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot {\color{blue}{y}}^{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot {y}^{2}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot {y}^{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 54.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 54.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+145}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+133}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+133) (+ 1.0 (* y (* x y))) (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+133) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = 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 (y <= 1.05d+133) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+133) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+133:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+133)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e+133)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+133], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e133

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 69.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 69.4%

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

   if 1.05e133 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 33.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 33.0%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 49.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 49.2%

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

Alternative 12: 58.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+116}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.2e+116) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+116) {
		tmp = 1.0;
	} else {
		tmp = 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 (y <= 5.2d+116) then
        tmp = 1.0d0
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+116) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e+116:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e+116)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e+116)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e+116], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999973e116

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   4. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{1} \]

   if 5.19999999999999973e116 < y

   1. Initial program 100.0%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 34.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 34.1%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 45.7%

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

Alternative 13: 52.0% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{\left(x \cdot y\right) \cdot y} \]
2. Add Preprocessing

4. Applied egg-rr 55.1%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))