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

Percentage Accurate: 100.0% → 100.0%

Time: 20.7s

Alternatives: 14

Speedup: 1.0×

• 24.6% 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: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+239}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2e+239)
     (exp x)
     (if (<= t_0 -5e+18)
       (* (pow x 3.0) 0.16666666666666666)
       (if (<= t_0 1e-13) (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 <= -2e+239) {
		tmp = exp(x);
	} else if (t_0 <= -5e+18) {
		tmp = pow(x, 3.0) * 0.16666666666666666;
	} else if (t_0 <= 1e-13) {
		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 <= -2e+239)
		tmp = exp(x);
	elseif (t_0 <= -5e+18)
		tmp = Float64((x ^ 3.0) * 0.16666666666666666);
	elseif (t_0 <= 1e-13)
		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, -2e+239], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, -5e+18], N[(N[Power[x, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[Exp[y], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -1.99999999999999998e239

   1. Initial program 100.0%

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

   4. Applied rewrites 62.5%

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

   if -1.99999999999999998e239 < (*.f64 (*.f64 x y) y) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 32.2%

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

   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 2.7

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

   7. Applied rewrites 2.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 48.7%

       \[\leadsto {x}^{3} \cdot \color{blue}{0.16666666666666666} \]

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1e-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 44.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2 \cdot 10^{+239}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (exp x)
     (if (<= t_0 1e-13) (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 <= -5e+18) {
		tmp = exp(x);
	} else if (t_0 <= 1e-13) {
		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 <= -5e+18)
		tmp = exp(x);
	elseif (t_0 <= 1e-13)
		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, -5e+18], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e-13], 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) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1e-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 44.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (exp x)
     (if (<= t_0 2e-60)
       (fma (* y x) y 1.0)
       (fma
        (fma (* (fma (* y y) (* 0.16666666666666666 x) 0.5) (* x x)) (* y y) x)
        (* y y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = exp(x);
	} else if (t_0 <= 2e-60) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma((fma((y * y), (0.16666666666666666 * x), 0.5) * (x * x)), (y * y), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -5e+18)
		tmp = exp(x);
	elseif (t_0 <= 2e-60)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(Float64(fma(Float64(y * y), Float64(0.16666666666666666 * x), 0.5) * Float64(x * x)), Float64(y * y), x), Float64(y * 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, -5e+18], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2e-60], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1.9999999999999999e-60

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1.9999999999999999e-60 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.1%

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

   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 76.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 2e-60)
       (fma (* y x) y 1.0)
       (fma
        (fma (* (fma (* y y) (* 0.16666666666666666 x) 0.5) (* x x)) (* y y) x)
        (* y y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e-60) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma((fma((y * y), (0.16666666666666666 * x), 0.5) * (x * x)), (y * y), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e-60)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(Float64(fma(Float64(y * y), Float64(0.16666666666666666 * x), 0.5) * Float64(x * x)), Float64(y * y), x), Float64(y * 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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-60], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1.9999999999999999e-60

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1.9999999999999999e-60 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.1%

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

   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 76.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 5000000.0)
       (fma (* y x) y 1.0)
       (if (<= t_0 1e+306) (* (fma 0.16666666666666666 x 0.5) (* x x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 5000000.0) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 1e+306) {
		tmp = fma(0.16666666666666666, x, 0.5) * (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 5000000.0)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 1e+306)
		tmp = Float64(fma(0.16666666666666666, x, 0.5) * Float64(x * x));
	else
		tmp = t_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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5000000.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+306], N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 5e6

   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 5e6 < (*.f64 (*.f64 x y) y) < 1.00000000000000002e306

   1. Initial program 100.0%

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

   4. Applied rewrites 50.2%

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

   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 39.2

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

   7. Applied rewrites 39.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 38.6%

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

   if 1.00000000000000002e306 < (*.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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -5e+18)
     t_1
     (if (<= t_0 5000000.0) 1.0 (if (<= t_0 1e+292) t_1 t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		tmp = 1.0;
	} else if (t_0 <= 1e+292) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = (x * x) * 0.5d0
    if (t_0 <= (-5d+18)) then
        tmp = t_1
    else if (t_0 <= 5000000.0d0) then
        tmp = 1.0d0
    else if (t_0 <= 1d+292) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		tmp = 1.0;
	} else if (t_0 <= 1e+292) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = (x * x) * 0.5
	tmp = 0
	if t_0 <= -5e+18:
		tmp = t_1
	elif t_0 <= 5000000.0:
		tmp = 1.0
	elif t_0 <= 1e+292:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(Float64(x * x) * 0.5)
	tmp = 0.0
	if (t_0 <= -5e+18)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		tmp = 1.0;
	elseif (t_0 <= 1e+292)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	t_1 = (x * x) * 0.5;
	tmp = 0.0;
	if (t_0 <= -5e+18)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		tmp = 1.0;
	elseif (t_0 <= 1e+292)
		tmp = t_1;
	else
		tmp = t_0;
	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[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+18], t$95$1, If[LessEqual[t$95$0, 5000000.0], 1.0, If[LessEqual[t$95$0, 1e+292], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18 or 5e6 < (*.f64 (*.f64 x y) y) < 1e292

   1. Initial program 100.0%

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

   4. Applied rewrites 48.0%

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

   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 12.1

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 26.3%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 5e6

   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 98.2%

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

   if 1e292 < (*.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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{+292}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 1e-13)
       (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 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e-13) {
		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 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 1e-13)
		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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], 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) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1e-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 44.5%

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

   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 31.5

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

   7. Applied rewrites 31.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 1e-13)
       (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 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e-13) {
		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 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 1e-13)
		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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], 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) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1e-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 44.5%

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

   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 31.5

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

   7. Applied rewrites 31.5%

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

   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 49.1%

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

   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 31.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 1e-13)
       (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 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e-13) {
		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 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 1e-13)
		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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], 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) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1e-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 44.5%

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

   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 31.5

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

   7. Applied rewrites 31.5%

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

   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 31.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18)
     (* (* x x) 0.5)
     (if (<= t_0 5e+22) (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 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 5e+22) {
		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 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 5e+22)
		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, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+22], 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) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 4.9999999999999996e22

   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 4.9999999999999996e22 < (*.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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+18) (* (* x x) 0.5) (if (<= t_0 1e-13) 1.0 (* (* y y) x)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e-13) {
		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 = (y * x) * y
    if (t_0 <= (-5d+18)) then
        tmp = (x * x) * 0.5d0
    else if (t_0 <= 1d-13) 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 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e-13) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if t_0 <= -5e+18:
		tmp = (x * x) * 0.5
	elif t_0 <= 1e-13:
		tmp = 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 <= -5e+18)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 1e-13)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -5e+18)
		tmp = (x * x) * 0.5;
	elseif (t_0 <= 1e-13)
		tmp = 1.0;
	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]}, If[LessEqual[t$95$0, -5e+18], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], 1.0, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18

   1. Initial program 100.0%

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

   4. Applied rewrites 47.6%

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

   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 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 25.1%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 1e-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 98.9%

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

   if 1e-13 < (*.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 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -5e+18) t_1 (if (<= t_0 5000000.0) 1.0 t_1))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		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 = (x * x) * 0.5d0
    if (t_0 <= (-5d+18)) then
        tmp = t_1
    else if (t_0 <= 5000000.0d0) 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 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+18) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = (x * x) * 0.5
	tmp = 0
	if t_0 <= -5e+18:
		tmp = t_1
	elif t_0 <= 5000000.0:
		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(x * x) * 0.5)
	tmp = 0.0
	if (t_0 <= -5e+18)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		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 = (x * x) * 0.5;
	tmp = 0.0;
	if (t_0 <= -5e+18)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		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[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+18], t$95$1, If[LessEqual[t$95$0, 5000000.0], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5e18 or 5e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 54.9%

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

   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 20.4

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

   7. Applied rewrites 20.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 30.8%

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

   if -5e18 < (*.f64 (*.f64 x y) y) < 5e6

   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 98.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.9% 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 50.3%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))