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

Percentage Accurate: 100.0% → 100.0%

Time: 20.3s

Alternatives: 16

Speedup: 1.0×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000.0)
     (exp x)
     (if (<= t_0 4e+20) (fma (* y x) y 1.0) (exp y)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = exp(x);
	} else if (t_0 <= 4e+20) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = exp(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = exp(x);
	elseif (t_0 <= 4e+20)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = exp(y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 4e+20], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[Exp[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e3

   1. Initial program 100.0%

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

   4. Applied rewrites 63.1%

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

   if -2e3 < (*.f64 (*.f64 x y) y) < 4e20

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 4e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.1%

      \[\leadsto e^{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(y \cdot y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_1 \cdot 0.16666666666666666\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* y y) x)))
   (if (<= t_0 -2000.0)
     (exp x)
     (if (<= t_0 2e-13)
       (fma (* y x) y 1.0)
       (if (<= t_0 5e+221)
         (fma
          (fma (* (* t_1 0.16666666666666666) (* x x)) (* y y) x)
          (* y y)
          1.0)
         t_1)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (y * y) * x;
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = exp(x);
	} else if (t_0 <= 2e-13) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 5e+221) {
		tmp = fma(fma(((t_1 * 0.16666666666666666) * (x * x)), (y * y), x), (y * y), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(Float64(y * y) * x)
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = exp(x);
	elseif (t_0 <= 2e-13)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 5e+221)
		tmp = fma(fma(Float64(Float64(t_1 * 0.16666666666666666) * Float64(x * x)), Float64(y * y), x), Float64(y * y), 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -2000.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+221], N[(N[(N[(N[(t$95$1 * 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -2e3

   1. Initial program 100.0%

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

   4. Applied rewrites 63.1%

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

   if -2e3 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e221

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 15.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 65.6%

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

   if 5.0000000000000002e221 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(y \cdot y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_1 \cdot 0.16666666666666666\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* y y) x)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 2e-13)
       (fma (* y x) y 1.0)
       (if (<= t_0 5e+221)
         (fma
          (fma (* (* t_1 0.16666666666666666) (* x x)) (* y y) x)
          (* y y)
          1.0)
         t_1)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (y * y) * x;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 2e-13) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 5e+221) {
		tmp = fma(fma(((t_1 * 0.16666666666666666) * (x * x)), (y * y), x), (y * y), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(Float64(y * y) * x)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 2e-13)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 5e+221)
		tmp = fma(fma(Float64(Float64(t_1 * 0.16666666666666666) * Float64(x * x)), Float64(y * y), x), Float64(y * y), 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+221], N[(N[(N[(N[(t$95$1 * 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e221

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 15.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 65.6%

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

   if 5.0000000000000002e221 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+221}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, y, -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 4e+20)
       (fma (* y x) y 1.0)
       (/
        (* (fma 0.027777777777777776 (* y y) -0.25) (* y y))
        (fma 0.16666666666666666 y -0.5))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 4e+20) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (fma(0.027777777777777776, (y * y), -0.25) * (y * y)) / fma(0.16666666666666666, y, -0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 4e+20)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(fma(0.027777777777777776, Float64(y * y), -0.25) * Float64(y * y)) / fma(0.16666666666666666, y, -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+20], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(0.027777777777777776 * N[(y * y), $MachinePrecision] + -0.25), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 * y + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 4e20

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 4e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 47.6%

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

   12. Applied rewrites 49.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y}, -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, y, -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot x\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 2e-13)
       (fma (* y x) y 1.0)
       (if (<= t_0 1e+132)
         (fma (* (* 0.16666666666666666 x) x) x 1.0)
         (* (* y y) x))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 2e-13) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 1e+132) {
		tmp = fma(((0.16666666666666666 * x) * x), x, 1.0);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 2e-13)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 1e+132)
		tmp = fma(Float64(Float64(0.16666666666666666 * x) * x), x, 1.0);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+132], N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 9.99999999999999991e131

   1. Initial program 100.0%

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 46.2

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

   7. Applied rewrites 46.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 46.2%

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

   if 9.99999999999999991e131 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot x\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 2e-13)
       (fma (* y x) y 1.0)
       (if (<= t_0 1e+132) (fma (fma 0.5 x 1.0) x 1.0) (* (* y y) x))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 2e-13) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 1e+132) {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 2e-13)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 1e+132)
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+132], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 9.99999999999999991e131

   1. Initial program 100.0%

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 46.2

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

   7. Applied rewrites 46.2%

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

   if 9.99999999999999991e131 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* 0.5 (* x x))))
   (if (<= t_0 -2000000000000.0)
     t_1
     (if (<= t_0 2e-13)
       (fma (* y x) y 1.0)
       (if (<= t_0 1e+132) t_1 (* (* y y) x))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 1e+132) {
		tmp = t_1;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 1e+132)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], t$95$1, If[LessEqual[t$95$0, 2e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+132], t$95$1, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e12 or 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 9.99999999999999991e131

   1. Initial program 100.0%

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 12.2

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 24.5%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 9.99999999999999991e131 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* 0.5 (* x x))))
   (if (<= t_0 -2000000000000.0)
     t_1
     (if (<= t_0 2e-13) 1.0 (if (<= t_0 1e+132) t_1 (* (* y y) x))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else if (t_0 <= 1e+132) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * x) * y
    t_1 = 0.5d0 * (x * x)
    if (t_0 <= (-2000000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d-13) then
        tmp = 1.0d0
    else if (t_0 <= 1d+132) then
        tmp = t_1
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else if (t_0 <= 1e+132) {
		tmp = t_1;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = 0.5 * (x * x)
	tmp = 0
	if t_0 <= -2000000000000.0:
		tmp = t_1
	elif t_0 <= 2e-13:
		tmp = 1.0
	elif t_0 <= 1e+132:
		tmp = t_1
	else:
		tmp = (y * y) * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	elseif (t_0 <= 1e+132)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	t_1 = 0.5 * (x * x);
	tmp = 0.0;
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	elseif (t_0 <= 1e+132)
		tmp = t_1;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], t$95$1, If[LessEqual[t$95$0, 2e-13], 1.0, If[LessEqual[t$95$0, 1e+132], t$95$1, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e12 or 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 9.99999999999999991e131

   1. Initial program 100.0%

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 12.2

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 24.5%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 9.99999999999999991e131 < (*.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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* 0.5 (* x x))))
   (if (<= t_0 -2000000000000.0)
     t_1
     (if (<= t_0 2e-13) 1.0 (if (<= t_0 1e+132) t_1 (* (* 0.5 y) y))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else if (t_0 <= 1e+132) {
		tmp = t_1;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * x) * y
    t_1 = 0.5d0 * (x * x)
    if (t_0 <= (-2000000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d-13) then
        tmp = 1.0d0
    else if (t_0 <= 1d+132) then
        tmp = t_1
    else
        tmp = (0.5d0 * y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else if (t_0 <= 1e+132) {
		tmp = t_1;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = 0.5 * (x * x)
	tmp = 0
	if t_0 <= -2000000000000.0:
		tmp = t_1
	elif t_0 <= 2e-13:
		tmp = 1.0
	elif t_0 <= 1e+132:
		tmp = t_1
	else:
		tmp = (0.5 * y) * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	elseif (t_0 <= 1e+132)
		tmp = t_1;
	else
		tmp = Float64(Float64(0.5 * y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	t_1 = 0.5 * (x * x);
	tmp = 0.0;
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	elseif (t_0 <= 1e+132)
		tmp = t_1;
	else
		tmp = (0.5 * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], t$95$1, If[LessEqual[t$95$0, 2e-13], 1.0, If[LessEqual[t$95$0, 1e+132], t$95$1, N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e12 or 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y) < 9.99999999999999991e131

   1. Initial program 100.0%

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

   4. Applied rewrites 61.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 12.2

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 24.5%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

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

   if 9.99999999999999991e131 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 62.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 55.9

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

   7. Applied rewrites 55.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 55.9%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 64.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 4e+20)
       (fma (* y x) y 1.0)
       (fma (fma (fma 0.16666666666666666 y 0.5) y 1.0) y 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 4e+20) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 4e+20)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+20], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 4e20

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 4e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.7

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

   7. Applied rewrites 47.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 4e+20)
       (fma (* y x) y 1.0)
       (fma (* (* y y) 0.16666666666666666) y 1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 4e+20) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(((y * y) * 0.16666666666666666), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 4e+20)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(Float64(Float64(y * y) * 0.16666666666666666), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+20], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 4e20

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 4e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.0%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 47.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2000000000000.0)
     (* 0.5 (* x x))
     (if (<= t_0 4e+20)
       (fma (* y x) y 1.0)
       (* (* (fma 0.16666666666666666 y 0.5) y) y)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = 0.5 * (x * x);
	} else if (t_0 <= 4e+20) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (fma(0.16666666666666666, y, 0.5) * y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = Float64(0.5 * Float64(x * x));
	elseif (t_0 <= 4e+20)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, y, 0.5) * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+20], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 18.3%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 4e20

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 4e20 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.7

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 47.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.28e-98)
   (fma (* y x) y 1.0)
   (if (<= y 9.4e+114)
     (fma (fma (* 0.5 (* x x)) (* y y) x) (* y y) 1.0)
     (* (* (fma 0.16666666666666666 y 0.5) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.28e-98) {
		tmp = fma((y * x), y, 1.0);
	} else if (y <= 9.4e+114) {
		tmp = fma(fma((0.5 * (x * x)), (y * y), x), (y * y), 1.0);
	} else {
		tmp = (fma(0.16666666666666666, y, 0.5) * y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.28e-98)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (y <= 9.4e+114)
		tmp = fma(fma(Float64(0.5 * Float64(x * x)), Float64(y * y), x), Float64(y * y), 1.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, y, 0.5) * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.28e-98], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 9.4e+114], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.28e-98

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 1.28e-98 < y < 9.4000000000000001e114

   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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 7.3%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 61.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 60.0%

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

   if 9.4000000000000001e114 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 63.6

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

   7. Applied rewrites 63.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 63.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -2000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* 0.5 (* x x))))
   (if (<= t_0 -2000000000000.0) t_1 (if (<= t_0 2e-13) 1.0 t_1))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * x) * y
    t_1 = 0.5d0 * (x * x)
    if (t_0 <= (-2000000000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d-13) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = 0.5 * (x * x);
	double tmp;
	if (t_0 <= -2000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-13) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = 0.5 * (x * x)
	tmp = 0
	if t_0 <= -2000000000000.0:
		tmp = t_1
	elif t_0 <= 2e-13:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	t_1 = 0.5 * (x * x);
	tmp = 0.0;
	if (t_0 <= -2000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-13)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000000.0], t$95$1, If[LessEqual[t$95$0, 2e-13], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e12 or 2.0000000000000001e-13 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 63.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 17.6

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

   7. Applied rewrites 17.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.2%

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

   if -2e12 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.4% 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 x around 0

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

5. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))