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

Percentage Accurate: 100.0% → 100.0%

Time: 20.3s

Alternatives: 14

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, y \cdot y, 0.5\right) \cdot x, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* (* y x) y)) 0.0)
   (exp x)
   (fma
    (fma (* (* y y) x) (* (fma (* 0.16666666666666666 x) (* y y) 0.5) x) x)
    (* y y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp(((y * x) * y)) <= 0.0) {
		tmp = exp(x);
	} else {
		tmp = fma(fma(((y * y) * x), (fma((0.16666666666666666 * x), (y * y), 0.5) * x), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 0.0], N[Exp[x], $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 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 61.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.2%

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

   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 85.3%

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

   13. Applied rewrites 95.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, y \cdot y, 0.5\right) \cdot x, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 2e-32)
       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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e-32) {
		tmp = 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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e-32)
		tmp = 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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-32], 1.0, 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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2.00000000000000011e-32

   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 98.5%

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

   if 2.00000000000000011e-32 < (*.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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

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

   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 78.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666 \cdot x, 0.5\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 2e-32)
       1.0
       (fma
        (fma (* (* 0.16666666666666666 (* (* y y) x)) (* 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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e-32) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(((0.16666666666666666 * ((y * y) * x)) * (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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e-32)
		tmp = 1.0;
	else
		tmp = fma(fma(Float64(Float64(0.16666666666666666 * Float64(Float64(y * y) * x)) * 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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-32], 1.0, N[(N[(N[(N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]), $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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2.00000000000000011e-32

   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 98.5%

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

   if 2.00000000000000011e-32 < (*.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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

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

   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 78.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 77.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot x\right)\right) \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 2e+17)
       (fma (* y x) y 1.0)
       (if (<= t_0 5e+172)
         (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e+17) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 5e+172) {
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e+17)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 5e+172)
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+17], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+172], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2e17

   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%

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

   if 2e17 < (*.f64 (*.f64 x y) y) < 5.0000000000000001e172

   1. Initial program 100.0%

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

   4. Applied rewrites 70.9%

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

   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 42.9

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

   7. Applied rewrites 42.9%

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

   if 5.0000000000000001e172 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 52.2%

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

   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 49.0

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

   7. Applied rewrites 49.0%

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

   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 49.0%

       \[\leadsto \left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot \color{blue}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 2e-32)
       1.0
       (fma (fma (* 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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e-32) {
		tmp = 1.0;
	} else {
		tmp = fma(fma((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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e-32)
		tmp = 1.0;
	else
		tmp = fma(fma(Float64(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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e-32], 1.0, N[(N[(N[(0.5 * 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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2.00000000000000011e-32

   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 98.5%

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

   if 2.00000000000000011e-32 < (*.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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

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

   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 78.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 73.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(x \cdot x\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, y \cdot y, 0.5\right) \cdot x, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* y x) y) -5e+39)
   (* (* x x) 0.5)
   (fma
    (fma (* (* y y) x) (* (fma (* 0.16666666666666666 x) (* y y) 0.5) x) x)
    (* y y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= -5e+39) {
		tmp = (x * x) * 0.5;
	} else {
		tmp = fma(fma(((y * y) * x), (fma((0.16666666666666666 * x), (y * y), 0.5) * x), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= -5e+39)
		tmp = Float64(Float64(x * x) * 0.5);
	else
		tmp = fma(fma(Float64(Float64(y * y) * x), Float64(fma(Float64(0.16666666666666666 * x), Float64(y * y), 0.5) * x), x), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], -5e+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.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 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.0%

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

   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 84.4%

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

   13. Applied rewrites 94.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(0.16666666666666666 \cdot x, y \cdot y, 0.5\right) \cdot x, x\right), y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 0.001)
       (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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 0.001)
		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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.001], 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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 1e-3

   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 1e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 33.6

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

   7. Applied rewrites 33.6%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 0.001)
       (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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 0.001)
		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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.001], 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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 1e-3

   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 1e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 33.6

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

   7. Applied rewrites 33.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 33.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 2e+17) (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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 2e+17) {
		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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 2e+17)
		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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+17], 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) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2e17

   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%

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

   if 2e17 < (*.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 49.9

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

   5. Applied rewrites 49.9%

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

   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 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39) (* (* x x) 0.5) (if (<= t_0 0.001) 1.0 (* (* y y) x)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		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+39)) then
        tmp = (x * x) * 0.5d0
    else if (t_0 <= 0.001d0) 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+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		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+39:
		tmp = (x * x) * 0.5
	elif t_0 <= 0.001:
		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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 0.001)
		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+39)
		tmp = (x * x) * 0.5;
	elseif (t_0 <= 0.001)
		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+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.001], 1.0, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 1e-3

   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 97.8%

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

   if 1e-3 < (*.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 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 0.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -5e+39)
     (* (* x x) 0.5)
     (if (<= t_0 0.001) 1.0 (* (* 0.5 y) y)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * y
    if (t_0 <= (-5d+39)) then
        tmp = (x * x) * 0.5d0
    else if (t_0 <= 0.001d0) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -5e+39) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 0.001) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if t_0 <= -5e+39:
		tmp = (x * x) * 0.5
	elif t_0 <= 0.001:
		tmp = 1.0
	else:
		tmp = (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+39)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 0.001)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -5e+39)
		tmp = (x * x) * 0.5;
	elseif (t_0 <= 0.001)
		tmp = 1.0;
	else
		tmp = (0.5 * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+39], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.001], 1.0, N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000015e39

   1. Initial program 100.0%

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.3

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 20.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 1e-3

   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 97.8%

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

   if 1e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 33.6

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

   7. Applied rewrites 33.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 33.5%

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

   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 58.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 0.001:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -5e+39) t_1 (if (<= t_0 2e+17) 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+39) {
		tmp = t_1;
	} else if (t_0 <= 2e+17) {
		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+39)) then
        tmp = t_1
    else if (t_0 <= 2d+17) 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+39) {
		tmp = t_1;
	} else if (t_0 <= 2e+17) {
		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+39:
		tmp = t_1
	elif t_0 <= 2e+17:
		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+39)
		tmp = t_1;
	elseif (t_0 <= 2e+17)
		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+39)
		tmp = t_1;
	elseif (t_0 <= 2e+17)
		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+39], t$95$1, If[LessEqual[t$95$0, 2e+17], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000015e39 or 2e17 < (*.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 17.9

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

   7. Applied rewrites 17.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 27.4%

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

   if -5.00000000000000015e39 < (*.f64 (*.f64 x y) y) < 2e17

   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 97.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.5% 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 52.0%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))