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

Percentage Accurate: 100.0% → 100.0%

Time: 31.8s

Alternatives: 15

Speedup: 1.0×

• 25.0% 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: 75.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* y (* x y)))))
   (if (<= t_0 0.0)
     (* (* y (* x x)) (* y 0.5))
     (if (<= t_0 2.0)
       (fma x (* y y) 1.0)
       (* x (* x (* 0.5 (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = exp((y * (x * y)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y * (x * x)) * (y * 0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = x * (x * (0.5 * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(y * Float64(x * y)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * 0.5));
	elseif (t_0 <= 2.0)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.7

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

   7. Applied rewrites 1.7%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.8

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

   10. Applied rewrites 1.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 16.2

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

   12. Applied rewrites 16.2%

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

   if 0.0 < (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. lower-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. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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

   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. Applied rewrites 84.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 74.5%

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

Alternative 3: 66.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		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 (exp((y * (x * y))) <= 2.0d0) 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 (Math.exp((y * (x * y))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp((y * (x * y))) <= 2.0:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		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 (exp((y * (x * y))) <= 2.0)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

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

   4. Applied rewrites 66.4%

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

   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

   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. lower-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. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.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. lower-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

4. Final simplification 67.1%

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

Alternative 4: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -4e+14)
   (exp (* x y))
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -4e+14) {
		tmp = exp((x * y));
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -4e+14)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -4e+14], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   if -4e14 < (*.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. Applied rewrites 96.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -4e+14)
   (exp x)
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -4e+14) {
		tmp = exp(x);
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -4e+14)
		tmp = exp(x);
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -4e+14], N[Exp[x], $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 60.1%

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

   if -4e14 < (*.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. Applied rewrites 96.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -4e+14)
   (* y (* y (* y (* x (* 0.16666666666666666 (* x x))))))
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -4e+14) {
		tmp = y * (y * (y * (x * (0.16666666666666666 * (x * x)))));
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -4e+14)
		tmp = Float64(y * Float64(y * Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(x * x))))));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -4e+14], N[(y * N[(y * N[(y * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. cube-mult N/A

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 27.7

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

   10. Applied rewrites 27.7%

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

   if -4e14 < (*.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. Applied rewrites 96.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -4e+14)
     (* y (* y (* y (* x (* 0.16666666666666666 (* x x))))))
     (if (<= t_0 1e-14)
       (fma x (* y y) 1.0)
       (* x (* x (* 0.5 (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -4e+14) {
		tmp = y * (y * (y * (x * (0.16666666666666666 * (x * x)))));
	} else if (t_0 <= 1e-14) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = x * (x * (0.5 * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -4e+14)
		tmp = Float64(y * Float64(y * Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(x * x))))));
	elseif (t_0 <= 1e-14)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * 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, -4e+14], N[(y * N[(y * N[(y * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-14], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.5 * 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) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. cube-mult N/A

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 27.7

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

   10. Applied rewrites 27.7%

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

   if -4e14 < (*.f64 (*.f64 x y) y) < 9.99999999999999999e-15

   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. lower-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. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 9.99999999999999999e-15 < (*.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. Applied rewrites 84.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      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. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -4e+14)
     (* (* y (* x x)) (* y 0.5))
     (if (<= t_0 4e+16)
       (fma x (* y y) 1.0)
       (fma x (fma x (* (* y y) 0.5) y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -4e+14) {
		tmp = (y * (x * x)) * (y * 0.5);
	} else if (t_0 <= 4e+16) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = fma(x, fma(x, ((y * y) * 0.5), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -4e+14)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * 0.5));
	elseif (t_0 <= 4e+16)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = fma(x, fma(x, Float64(Float64(y * y) * 0.5), 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, -4e+14], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+16], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.7

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

   7. Applied rewrites 1.7%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.8

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

   10. Applied rewrites 1.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 16.2

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

   12. Applied rewrites 16.2%

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

   if -4e14 < (*.f64 (*.f64 x y) y) < 4e16

   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. lower-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. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4e16 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.7%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot 0.5\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -4e+14)
     (* (* y (* x x)) (* y 0.5))
     (if (<= t_0 4e+16) (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 <= -4e+14) {
		tmp = (y * (x * x)) * (y * 0.5);
	} else if (t_0 <= 4e+16) {
		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 <= -4e+14)
		tmp = Float64(Float64(y * Float64(x * x)) * Float64(y * 0.5));
	elseif (t_0 <= 4e+16)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(Float64(y * 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, -4e+14], N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+16], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.7

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

   7. Applied rewrites 1.7%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 1.8

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

   10. Applied rewrites 1.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 16.2

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

   12. Applied rewrites 16.2%

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

   if -4e14 < (*.f64 (*.f64 x y) y) < 4e16

   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. lower-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. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4e16 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.7%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.6

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

   10. Applied rewrites 82.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\left(y \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot 0.5\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x, \left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5, x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -4e+14)
   (* y (* y (* y (* x (* 0.16666666666666666 (* x x))))))
   (fma (* y y) (fma x (* (* x (* y y)) 0.5) x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -4e+14) {
		tmp = y * (y * (y * (x * (0.16666666666666666 * (x * x)))));
	} else {
		tmp = fma((y * y), fma(x, ((x * (y * y)) * 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -4e+14)
		tmp = Float64(y * Float64(y * Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(x * x))))));
	else
		tmp = fma(Float64(y * y), fma(x, Float64(Float64(x * Float64(y * y)) * 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -4e+14], N[(y * N[(y * N[(y * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e14

   1. Initial program 100.0%

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 1.6

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. cube-mult N/A

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 27.7

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

   10. Applied rewrites 27.7%

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

   if -4e14 < (*.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. Applied rewrites 95.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -4 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x, \left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5, x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 4e+16) 1.0 (* x (* x (* (* y y) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 4e+16) {
		tmp = 1.0;
	} else {
		tmp = x * (x * ((y * y) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 4d+16) then
        tmp = 1.0d0
    else
        tmp = x * (x * ((y * y) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 4e+16) {
		tmp = 1.0;
	} else {
		tmp = x * (x * ((y * y) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 4e+16:
		tmp = 1.0
	else:
		tmp = x * (x * ((y * y) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 4e+16)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(Float64(y * y) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 4e+16)
		tmp = 1.0;
	else
		tmp = x * (x * ((y * y) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 4e+16], 1.0, N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 4e16

   1. Initial program 100.0%

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

   4. Applied rewrites 66.1%

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

   if 4e16 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.7%

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

   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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.6

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

   10. Applied rewrites 82.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e-14) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-14) {
		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)) <= 1e-14)
		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], 1e-14], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 9.99999999999999999e-15

   1. Initial program 100.0%

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

   4. Applied rewrites 66.4%

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

   if 9.99999999999999999e-15 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 43.9%

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

   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. lower-fma.f64 14.5

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

   7. Applied rewrites 14.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y (* x y)) 4e+16) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 4e+16) {
		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)) <= 4d+16) 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)) <= 4e+16) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 4e+16:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 4e+16)
		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)) <= 4e+16)
		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], 4e+16], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 4e16

   1. Initial program 100.0%

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

   4. Applied rewrites 66.1%

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

   if 4e16 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.7%

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

   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. lower-fma.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 4 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.3% 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. lower-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. lower-*.f64 67.0

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

5. Applied rewrites 67.0%

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

Alternative 15: 51.1% 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 rewrites 52.6%

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

Reproduce

herbie shell --seed 2024212 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))