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

Percentage Accurate: 100.0% → 100.0%

Time: 30.4s

Alternatives: 17

Speedup: 1.0×

• 24.4% 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: 69.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((y * x) * y))
    if (t_0 <= 0.0d0) then
        tmp = (0.5d0 * x) * x
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = math.exp(((y * x) * y))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (0.5 * x) * x
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = (y * y) * x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp(((y * x) * y));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (0.5 * x) * x;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 58.2%

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

   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 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 11.4%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 97.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.4%

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

4. Final simplification 69.8%

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

Alternative 3: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (exp (* y x))
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (if (<= t_0 2e+58) (exp x) (exp y))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = exp((y * x));
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 2e+58) {
		tmp = exp(x);
	} else {
		tmp = exp(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = exp(Float64(y * x));
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 2e+58)
		tmp = exp(x);
	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, -1.0], N[Exp[N[(y * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+58], N[Exp[x], $MachinePrecision], N[Exp[y], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 49.6%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 2e6 < (*.f64 (*.f64 x y) y) < 1.99999999999999989e58

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 50.9%

      \[\leadsto e^{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+58}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (exp x)
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (if (<= t_0 2e+58) (exp x) (exp y))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = exp(x);
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 2e+58) {
		tmp = exp(x);
	} else {
		tmp = exp(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = exp(x);
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 2e+58)
		tmp = exp(x);
	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, -1.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+58], N[Exp[x], $MachinePrecision], N[Exp[y], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -1 or 2e6 < (*.f64 (*.f64 x y) y) < 1.99999999999999989e58

   1. Initial program 99.9%

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

   4. Applied rewrites 61.2%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 50.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+58}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (exp x)
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (fma
        (fma (* (fma (* 0.16666666666666666 y) x 0.5) (* y y)) x y)
        x
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = exp(x);
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma((fma((0.16666666666666666 * y), x, 0.5) * (y * y)), x, y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = exp(x);
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(Float64(fma(Float64(0.16666666666666666 * y), x, 0.5) * Float64(y * y)), x, y), x, 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, -1.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 56.6%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 2e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 48.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (* (* 0.5 x) x)
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (if (<= t_0 2e+58)
         (* (* (fma x 0.16666666666666666 0.5) x) x)
         (if (<= t_0 2e+176)
           (fma (fma (fma 0.16666666666666666 y 0.5) y 1.0) y 1.0)
           (* (* x x) (* (* (fma 0.16666666666666666 (* y x) 0.5) y) y))))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 2e+58) {
		tmp = (fma(x, 0.16666666666666666, 0.5) * x) * x;
	} else if (t_0 <= 2e+176) {
		tmp = fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0);
	} else {
		tmp = (x * x) * ((fma(0.16666666666666666, (y * x), 0.5) * y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 2e+58)
		tmp = Float64(Float64(fma(x, 0.16666666666666666, 0.5) * x) * x);
	elseif (t_0 <= 2e+176)
		tmp = fma(fma(fma(0.16666666666666666, y, 0.5), y, 1.0), y, 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(fma(0.16666666666666666, Float64(y * x), 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, -1.0], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+58], N[(N[(N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2e+176], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * N[(y * x), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 56.6%

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

   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 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 11.1%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 2e6 < (*.f64 (*.f64 x y) y) < 1.99999999999999989e58

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   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 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.5%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 x y) y) < 2e176

   1. Initial program 100.0%

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

   4. Applied rewrites 54.3%

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

   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 35.0

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

   7. Applied rewrites 35.0%

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

   if 2e176 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 51.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 14.1

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

   7. Applied rewrites 14.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 51.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 48.8%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot y\right) \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot y\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (* (* 0.5 x) x)
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (fma
        (fma (* (fma (* 0.16666666666666666 y) x 0.5) (* y y)) x y)
        x
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma((fma((0.16666666666666666 * y), x, 0.5) * (y * y)), x, y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(Float64(fma(Float64(0.16666666666666666 * y), x, 0.5) * Float64(y * y)), x, y), x, 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, -1.0], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 56.6%

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

   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 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 11.1%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 2e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 48.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (* (* 0.5 x) x)
     (if (<= t_0 2000000.0)
       (fma (* y x) y 1.0)
       (if (<= t_0 2e+58)
         (* (* (fma x 0.16666666666666666 0.5) x) x)
         (* (* y y) x))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 2000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 2e+58) {
		tmp = (fma(x, 0.16666666666666666, 0.5) * x) * x;
	} 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 <= -1.0)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 2000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 2e+58)
		tmp = Float64(Float64(fma(x, 0.16666666666666666, 0.5) * x) * x);
	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, -1.0], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+58], N[(N[(N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 56.6%

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

   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 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 11.1%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 99.1

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

   5. Applied rewrites 99.1%

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

   if 2e6 < (*.f64 (*.f64 x y) y) < 1.99999999999999989e58

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   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 86.5

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.5%

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

   if 1.99999999999999989e58 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 73.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -1.0)
     (* (* 0.5 x) x)
     (if (<= t_0 2e+16) (fma (* y x) y 1.0) (* (* y y) x)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 2e+16) {
		tmp = fma((y * x), y, 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 <= -1.0)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 2e+16)
		tmp = fma(Float64(y * x), y, 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, -1.0], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2e+16], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   4. Applied rewrites 56.6%

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

   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 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 11.1%

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

   if -1 < (*.f64 (*.f64 x y) y) < 2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 98.3

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

   5. Applied rewrites 98.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(0.5 \cdot x\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* 0.5 x) x)))
   (if (<= t_0 -1e+56) t_1 (if (<= t_0 2000000.0) 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 <= -1e+56) {
		tmp = t_1;
	} else if (t_0 <= 2000000.0) {
		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 <= (-1d+56)) then
        tmp = t_1
    else if (t_0 <= 2000000.0d0) 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 <= -1e+56) {
		tmp = t_1;
	} else if (t_0 <= 2000000.0) {
		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 <= -1e+56:
		tmp = t_1
	elif t_0 <= 2000000.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(Float64(0.5 * x) * x)
	tmp = 0.0
	if (t_0 <= -1e+56)
		tmp = t_1;
	elseif (t_0 <= 2000000.0)
		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 <= -1e+56)
		tmp = t_1;
	elseif (t_0 <= 2000000.0)
		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[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+56], t$95$1, If[LessEqual[t$95$0, 2000000.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -1.00000000000000009e56 or 2e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 62.2%

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

   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 23.7

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 27.7%

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

   if -1.00000000000000009e56 < (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e-35)
   (fma (* y x) y 1.0)
   (if (<= y 3.4e+110)
     (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)
     (* (* (* 0.16666666666666666 y) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-35) {
		tmp = fma((y * x), y, 1.0);
	} else if (y <= 3.4e+110) {
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = ((0.16666666666666666 * y) * y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e-35)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (y <= 3.4e+110)
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e-35], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 3.4e+110], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.6999999999999999e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 76.5

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

   5. Applied rewrites 76.5%

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

   if 3.6999999999999999e-35 < y < 3.4000000000000001e110

   1. Initial program 100.0%

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

   4. Applied rewrites 71.9%

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

   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 46.5

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

   7. Applied rewrites 46.5%

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

   if 3.4000000000000001e110 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 48.2%

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

   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 48.2

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

   7. Applied rewrites 48.2%

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

   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 48.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 48.2%

       \[\leadsto \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 65.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot x\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e-35)
   (fma (* y x) y 1.0)
   (if (<= y 3.4e+110)
     (fma (* (* 0.16666666666666666 x) x) x 1.0)
     (* (* (* 0.16666666666666666 y) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-35) {
		tmp = fma((y * x), y, 1.0);
	} else if (y <= 3.4e+110) {
		tmp = fma(((0.16666666666666666 * x) * x), x, 1.0);
	} else {
		tmp = ((0.16666666666666666 * y) * y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e-35)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (y <= 3.4e+110)
		tmp = fma(Float64(Float64(0.16666666666666666 * x) * x), x, 1.0);
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e-35], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 3.4e+110], N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.6999999999999999e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 76.5

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

   5. Applied rewrites 76.5%

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

   if 3.6999999999999999e-35 < y < 3.4000000000000001e110

   1. Initial program 100.0%

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

   4. Applied rewrites 71.9%

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

   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 46.5

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

   7. Applied rewrites 46.5%

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

   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 46.5%

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

   if 3.4000000000000001e110 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 48.2%

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

   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 48.2

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

   7. Applied rewrites 48.2%

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

   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 48.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 48.2%

       \[\leadsto \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot x\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-34)
   (fma (* y x) y 1.0)
   (if (<= y 3.4e+110)
     (fma (fma 0.5 x 1.0) x 1.0)
     (* (* (* 0.16666666666666666 y) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-34) {
		tmp = fma((y * x), y, 1.0);
	} else if (y <= 3.4e+110) {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	} else {
		tmp = ((0.16666666666666666 * y) * y) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-34)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (y <= 3.4e+110)
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * y) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e-34], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 3.4e+110], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.19999999999999996e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 76.5

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

   5. Applied rewrites 76.5%

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

   if 1.19999999999999996e-34 < y < 3.4000000000000001e110

   1. Initial program 100.0%

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

   4. Applied rewrites 71.9%

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

   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 40.6

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

   7. Applied rewrites 40.6%

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

   if 3.4000000000000001e110 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 48.2%

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

   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 48.2

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

   7. Applied rewrites 48.2%

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

   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 48.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 48.2%

       \[\leadsto \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+157}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+58) 1.0 (if (<= y 4.8e+157) (* (* 0.5 x) x) (* (* 0.5 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+58) {
		tmp = 1.0;
	} else if (y <= 4.8e+157) {
		tmp = (0.5 * x) * x;
	} 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) :: tmp
    if (y <= 7.2d+58) then
        tmp = 1.0d0
    else if (y <= 4.8d+157) then
        tmp = (0.5d0 * x) * x
    else
        tmp = (0.5d0 * y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+58) {
		tmp = 1.0;
	} else if (y <= 4.8e+157) {
		tmp = (0.5 * x) * x;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+58:
		tmp = 1.0
	elif y <= 4.8e+157:
		tmp = (0.5 * x) * x
	else:
		tmp = (0.5 * y) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+58)
		tmp = 1.0;
	elseif (y <= 4.8e+157)
		tmp = Float64(Float64(0.5 * x) * x);
	else
		tmp = Float64(Float64(0.5 * y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e+58)
		tmp = 1.0;
	elseif (y <= 4.8e+157)
		tmp = (0.5 * x) * x;
	else
		tmp = (0.5 * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+58], 1.0, If[LessEqual[y, 4.8e+157], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.19999999999999993e58

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.0%

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

   if 7.19999999999999993e58 < y < 4.7999999999999999e157

   1. Initial program 100.0%

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

   4. Applied rewrites 51.6%

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

   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 28.9

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

   7. Applied rewrites 28.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 33.0%

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

   if 4.7999999999999999e157 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 57.3%

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

   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 57.3

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

   7. Applied rewrites 57.3%

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

   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 57.3%

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

   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 57.3%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+157}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) 5e-6) 1.0 (fma y x 1.0)))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], 5e-6], 1.0, N[(y * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 68.8%

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

   if 5.00000000000000041e-6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.1

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

   7. Applied rewrites 10.1%

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

4. Final simplification 53.6%

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

Alternative 16: 53.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) 2000000.0) 1.0 (* y x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 2000000.0) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y * x) * y) <= 2000000.0d0) then
        tmp = 1.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 2000000.0) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * x) * y) <= 2000000.0:
		tmp = 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= 2000000.0)
		tmp = 1.0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * x) * y) <= 2000000.0)
		tmp = 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], 2000000.0], 1.0, N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 68.4%

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

   if 2e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 10.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 2000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.0% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))