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

Percentage Accurate: 100.0% → 100.0%

Time: 27.5s

Alternatives: 18

Speedup: 1.0×

• 25.2% 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

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

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(Float64(y * x) * y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 70.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 69.7%

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

   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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 86.8%

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

4. Final simplification 75.0%

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

Alternative 3: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* (* y x) y)) 2.0)
   (fma (* y x) y 1.0)
   (* (* (* (* (* (* y y) x) x) x) y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (exp(((y * x) * y)) <= 2.0) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (((((y * y) * x) * x) * x) * y) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(Float64(y * x) * y)) <= 2.0)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y * y) * x) * x) * x) * y) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 69.7%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 16.6

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 53.5%

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

4. Final simplification 64.7%

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

Alternative 4: 65.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 69.7%

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

   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 y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.1%

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

4. Final simplification 68.3%

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

Alternative 5: 65.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.4%

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

   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 y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.1%

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

4. Final simplification 68.1%

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

Alternative 6: 53.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.4%

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

   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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 16.6

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

   7. Applied rewrites 16.6%

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

4. Final simplification 53.1%

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

Alternative 7: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (exp (* y x))
   (fma
    (*
     (* (fma (* (fma (* y x) (* 0.16666666666666666 y) 0.5) x) (* y y) 1.0) y)
     y)
    x
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = exp((y * x));
	} else {
		tmp = fma(((fma((fma((y * x), (0.16666666666666666 * y), 0.5) * x), (y * y), 1.0) * y) * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = exp(Float64(y * x));
	else
		tmp = fma(Float64(Float64(fma(Float64(fma(Float64(y * x), Float64(0.16666666666666666 * y), 0.5) * x), Float64(y * y), 1.0) * y) * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[Exp[N[(y * x), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(y * x), $MachinePrecision] * N[(0.16666666666666666 * y), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 38.2%

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

   if -5e12 < (*.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 y around 0

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (exp x)
   (fma
    (*
     (* (fma (* (fma (* y x) (* 0.16666666666666666 y) 0.5) x) (* y y) 1.0) y)
     y)
    x
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = exp(x);
	} else {
		tmp = fma(((fma((fma((y * x), (0.16666666666666666 * y), 0.5) * x), (y * y), 1.0) * y) * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = exp(x);
	else
		tmp = fma(Float64(Float64(fma(Float64(fma(Float64(y * x), Float64(0.16666666666666666 * y), 0.5) * x), Float64(y * y), 1.0) * y) * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[Exp[x], $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(y * x), $MachinePrecision] * N[(0.16666666666666666 * y), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 64.1%

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

   if -5e12 < (*.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 y around 0

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (* (* (* (* (fma 0.16666666666666666 x (/ 0.5 y)) (* x x)) y) y) y)
   (fma
    (*
     (* (fma (* (fma (* y x) (* 0.16666666666666666 y) 0.5) x) (* y y) 1.0) y)
     y)
    x
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = (((fma(0.16666666666666666, x, (0.5 / y)) * (x * x)) * y) * y) * y;
	} else {
		tmp = fma(((fma((fma((y * x), (0.16666666666666666 * y), 0.5) * x), (y * y), 1.0) * y) * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = Float64(Float64(Float64(Float64(fma(0.16666666666666666, x, Float64(0.5 / y)) * Float64(x * x)) * y) * y) * y);
	else
		tmp = fma(Float64(Float64(fma(Float64(fma(Float64(y * x), Float64(0.16666666666666666 * y), 0.5) * x), Float64(y * y), 1.0) * y) * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * x + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(y * x), $MachinePrecision] * N[(0.16666666666666666 * y), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 2.2

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 18.0%

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

   if -5e12 < (*.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 y around 0

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot x, 0.16666666666666666 \cdot y, 0.5\right) \cdot x, y \cdot y, 1\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (* (* (* (* (fma 0.16666666666666666 x (/ 0.5 y)) (* x x)) y) y) y)
   (fma
    (* (* (* (* (* (* (* (* y y) x) x) 0.16666666666666666) y) y) y) y)
    x
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = (((fma(0.16666666666666666, x, (0.5 / y)) * (x * x)) * y) * y) * y;
	} else {
		tmp = fma(((((((((y * y) * x) * x) * 0.16666666666666666) * y) * y) * y) * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = Float64(Float64(Float64(Float64(fma(0.16666666666666666, x, Float64(0.5 / y)) * Float64(x * x)) * y) * y) * y);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y * y) * x) * x) * 0.16666666666666666) * y) * y) * y) * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * x + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 2.2

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 18.0%

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

   if -5e12 < (*.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 y around 0

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 97.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot y, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y, y \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (* (* (* (* (fma 0.16666666666666666 x (/ 0.5 y)) (* x x)) y) y) y)
   (fma
    (* (* (* (* (* (* (* y y) x) x) 0.16666666666666666) y) y) y)
    (* y x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = (((fma(0.16666666666666666, x, (0.5 / y)) * (x * x)) * y) * y) * y;
	} else {
		tmp = fma((((((((y * y) * x) * x) * 0.16666666666666666) * y) * y) * y), (y * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = Float64(Float64(Float64(Float64(fma(0.16666666666666666, x, Float64(0.5 / y)) * Float64(x * x)) * y) * y) * y);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y * y) * x) * x) * 0.16666666666666666) * y) * y) * y), Float64(y * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * x + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 2.2

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 18.0%

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

   if -5e12 < (*.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 y around 0

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 94.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y, y \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x, 0.5, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5000000000000.0)
   (* (* (* (* (fma 0.16666666666666666 x (/ 0.5 y)) (* x x)) y) y) y)
   (fma (fma (* (* (* y y) x) x) 0.5 x) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5000000000000.0) {
		tmp = (((fma(0.16666666666666666, x, (0.5 / y)) * (x * x)) * y) * y) * y;
	} else {
		tmp = fma(fma((((y * y) * x) * x), 0.5, x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5000000000000.0)
		tmp = Float64(Float64(Float64(Float64(fma(0.16666666666666666, x, Float64(0.5 / y)) * Float64(x * x)) * y) * y) * y);
	else
		tmp = fma(fma(Float64(Float64(Float64(y * y) * x) * x), 0.5, x), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * x + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e12

   1. Initial program 100.0%

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 2.2

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

   7. Applied rewrites 2.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 1.6%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 18.0%

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

   if -5e12 < (*.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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5000000000000:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, x, \frac{0.5}{y}\right) \cdot \left(x \cdot x\right)\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x, 0.5, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.5\right) \cdot y\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) 1e+15)
   (fma (* y x) y 1.0)
   (* (* (* (* (* (* y y) y) 0.5) y) x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 1e+15) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (((((y * y) * y) * 0.5) * y) * x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= 1e+15)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y * y) * y) * 0.5) * y) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], 1e+15], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e15

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 69.3%

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

   if 1e15 < (*.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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 87.9%

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

   12. Applied rewrites 89.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.5\right) \cdot y\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x, 0.5, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.5\right) \cdot x\right) \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.12e+98)
   (fma (fma (* (* (* y y) x) x) 0.5 x) (* y y) 1.0)
   (* (* (* (* (* (* y y) y) 0.5) x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.12e+98) {
		tmp = fma(fma((((y * y) * x) * x), 0.5, x), (y * y), 1.0);
	} else {
		tmp = (((((y * y) * y) * 0.5) * x) * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.12e+98)
		tmp = fma(fma(Float64(Float64(Float64(y * y) * x) * x), 0.5, x), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y * y) * y) * 0.5) * x) * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.12e+98], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e98

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if 1.12e98 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 63.1%

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

Alternative 15: 65.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, 0.5, x\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-120)
   (fma (* y x) y 1.0)
   (if (<= y 1.95e+80)
     (fma (fma (* (* x x) y) 0.5 x) y 1.0)
     (fma (* y y) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-120) {
		tmp = fma((y * x), y, 1.0);
	} else if (y <= 1.95e+80) {
		tmp = fma(fma(((x * x) * y), 0.5, x), y, 1.0);
	} else {
		tmp = fma((y * y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.55e-120)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (y <= 1.95e+80)
		tmp = fma(fma(Float64(Float64(x * x) * y), 0.5, x), y, 1.0);
	else
		tmp = fma(Float64(y * y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.55e-120], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 1.95e+80], N[(N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * 0.5 + x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5500000000000001e-120

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 71.0%

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

   if 1.5500000000000001e-120 < y < 1.94999999999999999e80

   1. Initial program 100.0%

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if 1.94999999999999999e80 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

Alternative 16: 53.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* (* y x) y) 5e+24) 1.0 (* y x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 5e+24) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	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) <= 5d+24) then
        tmp = 1.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 5e+24) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * x) * y) <= 5e+24:
		tmp = 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= 5e+24)
		tmp = 1.0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * x) * y) <= 5e+24)
		tmp = 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], 5e+24], 1.0, N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.3%

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

   if 5.00000000000000045e24 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in y 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. *-commutative N/A

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

      3. lower-fma.f64 17.2

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 16.9%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot y, x, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* y y) x 1.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 68.3

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

5. Applied rewrites 68.3%

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

Alternative 18: 50.7% 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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 48.9%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))