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

Percentage Accurate: 100.0% → 100.0%

Time: 34.2s

Alternatives: 16

Speedup: 1.0×

• 24.8% 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: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2.0)
   (fma (* x y) y 1.0)
   (* x (* x (* y (* y 0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = x * (x * (y * (y * 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(y * Float64(y * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(x * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 2

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 67.2

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

   7. Applied egg-rr 67.2%

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

   if 2 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 47.4

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

   7. Simplified 47.4%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 47.4%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. lft-mult-inverse N/A

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

       5. associate-*l* N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. associate-*l/ N/A

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

       9. associate-/l* N/A

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

       10. associate-*l/ N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

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

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. associate-*l/ N/A

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

       17. associate-/l* N/A

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

       18. associate-*l/ N/A

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

       19. metadata-eval N/A

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

       20. associate-*r/ N/A

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

       21. associate-*l* N/A

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

       22. lft-mult-inverse N/A

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

       23. metadata-eval N/A

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

       24. unpow2 N/A

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

       25. associate-*l* N/A

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

       26. *-commutative N/A

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

   13. Simplified 76.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -2e+20)
     (exp (* x y))
     (fma t_0 (fma t_0 (fma x (* (* y y) 0.16666666666666666) 0.5) 1.0) 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -2e+20) {
		tmp = exp((x * y));
	} else {
		tmp = fma(t_0, fma(t_0, fma(x, ((y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -2e+20)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(t_0, fma(t_0, fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -2e+20], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e20

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.3%

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

   if -2e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -2e+20)
     (exp x)
     (fma t_0 (fma t_0 (fma x (* (* y y) 0.16666666666666666) 0.5) 1.0) 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -2e+20) {
		tmp = exp(x);
	} else {
		tmp = fma(t_0, fma(t_0, fma(x, ((y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -2e+20)
		tmp = exp(x);
	else
		tmp = fma(t_0, fma(t_0, fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -2e+20], N[Exp[x], $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e20

   1. Initial program 100.0%

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

   4. Applied egg-rr 60.6%

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

   if -2e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+112)
     (* (* y (* x x)) (* y (fma x (* y 0.16666666666666666) 0.5)))
     (if (<= t_0 5e+27)
       (fma (* x (* y y)) (fma x (* (* y y) 0.5) 1.0) 1.0)
       (* x (* 0.5 (* x (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+112) {
		tmp = (y * (x * x)) * (y * fma(x, (y * 0.16666666666666666), 0.5));
	} else if (t_0 <= 5e+27) {
		tmp = fma((x * (y * y)), fma(x, ((y * y) * 0.5), 1.0), 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+112)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * fma(x, Float64(y * 0.16666666666666666), 0.5)));
	elseif (t_0 <= 5e+27)
		tmp = fma(Float64(x * Float64(y * y)), fma(x, Float64(Float64(y * y) * 0.5), 1.0), 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+112], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+27], N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 1.6

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

   7. Simplified 1.6%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 1.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 18.0

          \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot 0.16666666666666666}, 0.5\right)\right) \]

   12. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)} \]

   if -5e112 < (*.f64 (*.f64 x y) y) < 4.99999999999999979e27

   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

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

   5. Simplified 91.3%

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

   if 4.99999999999999979e27 < (*.f64 (*.f64 x y) 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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x 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

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.2

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

   8. Simplified 89.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -5e+112)
     (* (* y (* x x)) (* y (fma x (* y 0.16666666666666666) 0.5)))
     (fma t_0 (fma t_0 (fma x (* (* y y) 0.16666666666666666) 0.5) 1.0) 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -5e+112) {
		tmp = (y * (x * x)) * (y * fma(x, (y * 0.16666666666666666), 0.5));
	} else {
		tmp = fma(t_0, fma(t_0, fma(x, ((y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -5e+112)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * fma(x, Float64(y * 0.16666666666666666), 0.5)));
	else
		tmp = fma(t_0, fma(t_0, fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -5e+112], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 1.6

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

   7. Simplified 1.6%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 1.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 18.0

          \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot 0.16666666666666666}, 0.5\right)\right) \]

   12. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)} \]

   if -5e112 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -5e+112)
     (* (* y (* x x)) (* y (fma x (* y 0.16666666666666666) 0.5)))
     (fma t_0 (fma t_0 (* x (* y (* y 0.16666666666666666))) 1.0) 1.0))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -5e+112) {
		tmp = (y * (x * x)) * (y * fma(x, (y * 0.16666666666666666), 0.5));
	} else {
		tmp = fma(t_0, fma(t_0, (x * (y * (y * 0.16666666666666666))), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -5e+112)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * fma(x, Float64(y * 0.16666666666666666), 0.5)));
	else
		tmp = fma(t_0, fma(t_0, Float64(x * Float64(y * Float64(y * 0.16666666666666666))), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -5e+112], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$0 * N[(x * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 1.6

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

   7. Simplified 1.6%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 1.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 18.0

          \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot 0.16666666666666666}, 0.5\right)\right) \]

   12. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)} \]

   if -5e112 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 91.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 90.9

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

   8. Simplified 90.9%

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)}, 1\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+112)
     (* (* y (* x x)) (* y (fma x (* y 0.16666666666666666) 0.5)))
     (if (<= t_0 5e+27)
       (fma (* x y) y 1.0)
       (* x (* 0.5 (* x (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+112) {
		tmp = (y * (x * x)) * (y * fma(x, (y * 0.16666666666666666), 0.5));
	} else if (t_0 <= 5e+27) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+112)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * fma(x, Float64(y * 0.16666666666666666), 0.5)));
	elseif (t_0 <= 5e+27)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+112], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+27], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 1.6

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

   7. Simplified 1.6%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 1.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       8. accelerator-lowering-fma.f64 N/A

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

       9. *-lowering-*.f64 18.0

          \[\leadsto \left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot 0.16666666666666666}, 0.5\right)\right) \]

   12. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)} \]

   if -5e112 < (*.f64 (*.f64 x y) y) < 4.99999999999999979e27

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 91.1

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

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 91.1

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

   7. Applied egg-rr 91.1%

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

   if 4.99999999999999979e27 < (*.f64 (*.f64 x y) 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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x 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

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.2

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

   8. Simplified 89.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+112)
     (* y (* 0.5 (* x (* x y))))
     (if (<= t_0 5e+27)
       (fma (* x y) y 1.0)
       (* x (* 0.5 (* x (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+112) {
		tmp = y * (0.5 * (x * (x * y)));
	} else if (t_0 <= 5e+27) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+112)
		tmp = Float64(y * Float64(0.5 * Float64(x * Float64(x * y))));
	elseif (t_0 <= 5e+27)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+112], N[(y * N[(0.5 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+27], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 2.0

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

   7. Simplified 2.0%

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

   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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 11.0

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

   10. Simplified 11.0%

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

   if -5e112 < (*.f64 (*.f64 x y) y) < 4.99999999999999979e27

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 91.1

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

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 91.1

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

   7. Applied egg-rr 91.1%

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

   if 4.99999999999999979e27 < (*.f64 (*.f64 x y) 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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in x 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

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.2

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

   8. Simplified 89.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(y \cdot y, 0.5, 0\right), y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+112)
     (* y (* 0.5 (* x (* x y))))
     (if (<= t_0 2e-6)
       (fma x (* y y) 1.0)
       (fma x (fma x (fma (* y y) 0.5 0.0) y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+112) {
		tmp = y * (0.5 * (x * (x * y)));
	} else if (t_0 <= 2e-6) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = fma(x, fma(x, fma((y * y), 0.5, 0.0), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+112)
		tmp = Float64(y * Float64(0.5 * Float64(x * Float64(x * y))));
	elseif (t_0 <= 2e-6)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = fma(x, fma(x, fma(Float64(y * y), 0.5, 0.0), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+112], N[(y * N[(0.5 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5 + 0.0), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 2.0

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

   7. Simplified 2.0%

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

   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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 11.0

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

   10. Simplified 11.0%

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

   if -5e112 < (*.f64 (*.f64 x y) y) < 1.99999999999999991e-6

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.4

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

   5. Simplified 92.4%

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

   if 1.99999999999999991e-6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 47.4

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

   7. Simplified 47.4%

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

   8. Taylor expanded in y around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot {y}^{2} + 0}, y\right), 1\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{{y}^{2} \cdot \frac{1}{2}} + 0, y\right), 1\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{1}{2}, 0\right)}, y\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{1}{2}, 0\right), y\right), 1\right) \]

      5. *-lowering-*.f64 77.0

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{y \cdot y}, 0.5, 0\right), y\right), 1\right) \]

   10. Simplified 77.0%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.5, 0\right)}, y\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(y \cdot y, 0.5, 0\right), y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+112)
     (* y (* 0.5 (* x (* x y))))
     (if (<= t_0 2e-6) (fma x (* y y) 1.0) (* x (* x (* y (* y 0.5))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+112) {
		tmp = y * (0.5 * (x * (x * y)));
	} else if (t_0 <= 2e-6) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = x * (x * (y * (y * 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+112)
		tmp = Float64(y * Float64(0.5 * Float64(x * Float64(x * y))));
	elseif (t_0 <= 2e-6)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(y * Float64(y * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+112], N[(y * N[(0.5 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e112

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.7%

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

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 2.0

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

   7. Simplified 2.0%

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

   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. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 11.0

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

   10. Simplified 11.0%

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

   if -5e112 < (*.f64 (*.f64 x y) y) < 1.99999999999999991e-6

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.4

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

   5. Simplified 92.4%

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

   if 1.99999999999999991e-6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 47.4

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

   7. Simplified 47.4%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l* N/A

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

   10. Simplified 47.4%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(x, y \cdot 0.16666666666666666, 0.5\right)\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. lft-mult-inverse N/A

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

       5. associate-*l* N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. associate-*l/ N/A

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

       9. associate-/l* N/A

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

       10. associate-*l/ N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

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

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. associate-*l/ N/A

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

       17. associate-/l* N/A

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

       18. associate-*l/ N/A

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

       19. metadata-eval N/A

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

       20. associate-*r/ N/A

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

       21. associate-*l* N/A

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

       22. lft-mult-inverse N/A

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

       23. metadata-eval N/A

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

       24. unpow2 N/A

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

       25. associate-*l* N/A

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

       26. *-commutative N/A

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

   13. Simplified 76.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2e-6) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e-6) {
		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 * (x * y)) <= 2d-6) 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 * (x * y)) <= 2e-6) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 2e-6:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e-6)
		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 * (x * y)) <= 2e-6)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e-6], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   4. Applied egg-rr 67.1%

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

   if 1.99999999999999991e-6 < (*.f64 (*.f64 x y) 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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 63.2

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

   8. Simplified 63.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2e-6) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e-6) {
		tmp = 1.0;
	} else {
		tmp = fma(x, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e-6)
		tmp = 1.0;
	else
		tmp = fma(x, y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e-6], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   4. Applied egg-rr 67.1%

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

   if 1.99999999999999991e-6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + 1} \]

      2. accelerator-lowering-fma.f64 13.8

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

   7. Simplified 13.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y (* x y)) 2e+64) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+64) {
		tmp = 1.0;
	} else {
		tmp = 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 * (x * y)) <= 2d+64) then
        tmp = 1.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+64) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 2e+64:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e+64)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 2e+64)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e+64], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2.00000000000000004e64

   1. Initial program 100.0%

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

   4. Applied egg-rr 64.9%

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

   if 2.00000000000000004e64 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + 1} \]

      2. accelerator-lowering-fma.f64 15.1

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

   7. Simplified 15.1%

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

   8. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 15.0

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

   10. Simplified 15.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (* y y) 1.0))

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 66.2

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

5. Simplified 66.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, 1\right)} \]
6. Add Preprocessing

Alternative 16: 51.6% accurate, 111.0× 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 51.6%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))