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

Percentage Accurate: 100.0% → 100.0%

Time: 19.9s

Alternatives: 16

Speedup: 1.0×

• 25.3% 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

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 69.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 5.0)
   (fma (* x y) y 1.0)
   (fma x (fma x (* 0.5 (* y y)) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 5.0) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = fma(x, fma(x, (0.5 * (y * y)), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 5.0)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = fma(x, fma(x, Float64(0.5 * Float64(y * y)), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5.0], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 5

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 65.1%

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

   if 5 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 69.7%

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

Alternative 3: 66.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.2%

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

4. Final simplification 67.2%

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

Alternative 4: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right), y \cdot \left(-y\right), x \cdot \left(x \cdot 0.5\right)\right), -x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -5e+138)
     (/ 1.0 (fma (* y y) (* x (fma x (* 0.5 (* y y)) -1.0)) 1.0))
     (if (<= t_0 -2e+19)
       (/
        1.0
        (fma
         (* y y)
         (fma
          (* y y)
          (fma
           (* x (* x (* x 0.16666666666666666)))
           (* y (- y))
           (* x (* x 0.5)))
          (- x))
         1.0))
       (fma
        (* y y)
        (fma (* x (* x (* y y))) (fma x (* 0.16666666666666666 (* y y)) 0.5) x)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -5e+138) {
		tmp = 1.0 / fma((y * y), (x * fma(x, (0.5 * (y * y)), -1.0)), 1.0);
	} else if (t_0 <= -2e+19) {
		tmp = 1.0 / fma((y * y), fma((y * y), fma((x * (x * (x * 0.16666666666666666))), (y * -y), (x * (x * 0.5))), -x), 1.0);
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, (0.16666666666666666 * (y * y)), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -5e+138)
		tmp = Float64(1.0 / fma(Float64(y * y), Float64(x * fma(x, Float64(0.5 * Float64(y * y)), -1.0)), 1.0));
	elseif (t_0 <= -2e+19)
		tmp = Float64(1.0 / fma(Float64(y * y), fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(x * 0.16666666666666666))), Float64(y * Float64(-y)), Float64(x * Float64(x * 0.5))), Float64(-x)), 1.0));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(0.16666666666666666 * Float64(y * y)), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+138], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e+19], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * (-y)), $MachinePrecision] + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -5.00000000000000016e138

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

   if -5.00000000000000016e138 < (*.f64 (*.f64 x y) y) < -2e19

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 1.9%

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

   7. Applied rewrites 1.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 56.3%

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

   if -2e19 < (*.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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 97.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right), y \cdot \left(-y\right), x \cdot \left(x \cdot 0.5\right)\right), -x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(-y\right), -x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -2e+19)
   (/
    1.0
    (fma
     (* y y)
     (fma
      y
      (*
       (fma 0.16666666666666666 (* y (* y (* x (* x x)))) (* (* x x) -0.5))
       (- y))
      (- x))
     1.0))
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* 0.16666666666666666 (* y y)) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -2e+19) {
		tmp = 1.0 / fma((y * y), fma(y, (fma(0.16666666666666666, (y * (y * (x * (x * x)))), ((x * x) * -0.5)) * -y), -x), 1.0);
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, (0.16666666666666666 * (y * y)), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -2e+19)
		tmp = Float64(1.0 / fma(Float64(y * y), fma(y, Float64(fma(0.16666666666666666, Float64(y * Float64(y * Float64(x * Float64(x * x)))), Float64(Float64(x * x) * -0.5)) * Float64(-y)), Float64(-x)), 1.0));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(0.16666666666666666 * Float64(y * y)), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -2e+19], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(0.16666666666666666 * N[(y * N[(y * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e19

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 93.1%

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

   if -2e19 < (*.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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 97.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, \mathsf{fma}\left(0.16666666666666666, y \cdot \left(y \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(-y\right), -x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot \left(x \cdot \left(x \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2e-202)
   (/ 1.0 (fma (* y y) (* x (fma x (* 0.5 (* y y)) -1.0)) 1.0))
   (fma
    (* y y)
    (fma (* y (* x (* x y))) (fma x (* 0.16666666666666666 (* y y)) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e-202) {
		tmp = 1.0 / fma((y * y), (x * fma(x, (0.5 * (y * y)), -1.0)), 1.0);
	} else {
		tmp = fma((y * y), fma((y * (x * (x * y))), fma(x, (0.16666666666666666 * (y * y)), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e-202)
		tmp = Float64(1.0 / fma(Float64(y * y), Float64(x * fma(x, Float64(0.5 * Float64(y * y)), -1.0)), 1.0));
	else
		tmp = fma(Float64(y * y), fma(Float64(y * Float64(x * Float64(x * y))), fma(x, Float64(0.16666666666666666 * Float64(y * y)), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e-202], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2.0000000000000001e-202

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 93.9%

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

   if 2.0000000000000001e-202 < (*.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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 96.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot \left(x \cdot \left(x \cdot y\right)\right), \mathsf{fma}\left(x, 0.16666666666666666 \cdot \left(y \cdot y\right), 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2.0)
   (/ 1.0 (fma (* y y) (* x (fma x (* 0.5 (* y y)) -1.0)) 1.0))
   (fma
    (* y y)
    (* (* y y) (* 0.16666666666666666 (* x (* x (* x (* y y))))))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2.0) {
		tmp = 1.0 / fma((y * y), (x * fma(x, (0.5 * (y * y)), -1.0)), 1.0);
	} else {
		tmp = fma((y * y), ((y * y) * (0.16666666666666666 * (x * (x * (x * (y * y)))))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2.0)
		tmp = Float64(1.0 / fma(Float64(y * y), Float64(x * fma(x, Float64(0.5 * Float64(y * y)), -1.0)), 1.0));
	else
		tmp = fma(Float64(y * y), Float64(Float64(y * y) * Float64(0.16666666666666666 * Float64(x * Float64(x * Float64(x * Float64(y * y)))))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * N[(x * N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 94.0%

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

   if 2 < (*.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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{6} \cdot \color{blue}{\left({x}^{3} \cdot {y}^{4}\right)}, 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.8%

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

4. Final simplification 94.7%

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

Alternative 8: 91.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 50000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, t\_0, -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (* y y))))
   (if (<= (* y (* x y)) 50000.0)
     (/ 1.0 (fma (* y y) (* x (fma x t_0 -1.0)) 1.0))
     (* (* x (* x y)) (* y t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (y * y);
	double tmp;
	if ((y * (x * y)) <= 50000.0) {
		tmp = 1.0 / fma((y * y), (x * fma(x, t_0, -1.0)), 1.0);
	} else {
		tmp = (x * (x * y)) * (y * t_0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 50000.0)
		tmp = Float64(1.0 / fma(Float64(y * y), Float64(x * fma(x, t_0, -1.0)), 1.0));
	else
		tmp = Float64(Float64(x * Float64(x * y)) * Float64(y * t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 50000.0], N[(1.0 / N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5e4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 93.5%

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

   if 5e4 < (*.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

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.9%

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

   10. Applied rewrites 95.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 50000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{-x \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -2e+19)
     (/ 1.0 (- (* x (* y y))))
     (if (<= t_0 2.0)
       (fma x (* y y) 1.0)
       (fma x (fma x (* 0.5 (* y y)) y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -2e+19) {
		tmp = 1.0 / -(x * (y * y));
	} else if (t_0 <= 2.0) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = fma(x, fma(x, (0.5 * (y * y)), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -2e+19)
		tmp = Float64(1.0 / Float64(-Float64(x * Float64(y * y))));
	elseif (t_0 <= 2.0)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = fma(x, fma(x, Float64(0.5 * Float64(y * y)), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+19], N[(1.0 / (-N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e19

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 67.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 67.8%

       \[\leadsto \frac{1}{-x \cdot \left(y \cdot y\right)} \]

   if -2e19 < (*.f64 (*.f64 x y) y) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{-x \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e-8)
   (/ 1.0 (- 1.0 (* x (* y y))))
   (* (* x (* x y)) (* y (* 0.5 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-8) {
		tmp = 1.0 / (1.0 - (x * (y * y)));
	} else {
		tmp = (x * (x * y)) * (y * (0.5 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 1d-8) then
        tmp = 1.0d0 / (1.0d0 - (x * (y * y)))
    else
        tmp = (x * (x * y)) * (y * (0.5d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-8) {
		tmp = 1.0 / (1.0 - (x * (y * y)));
	} else {
		tmp = (x * (x * y)) * (y * (0.5 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 1e-8:
		tmp = 1.0 / (1.0 - (x * (y * y)))
	else:
		tmp = (x * (x * y)) * (y * (0.5 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e-8)
		tmp = Float64(1.0 / Float64(1.0 - Float64(x * Float64(y * y))));
	else
		tmp = Float64(Float64(x * Float64(x * y)) * Float64(y * Float64(0.5 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 1e-8)
		tmp = 1.0 / (1.0 - (x * (y * y)));
	else
		tmp = (x * (x * y)) * (y * (0.5 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e-8], N[(1.0 / N[(1.0 - N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.4%

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

   if 1e-8 < (*.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

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.4%

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

   10. Applied rewrites 92.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) 1e-8)
     (/ 1.0 (- 1.0 t_0))
     (* x (* x (* 0.16666666666666666 (* y t_0)))))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= 1e-8) {
		tmp = 1.0 / (1.0 - t_0);
	} else {
		tmp = x * (x * (0.16666666666666666 * (y * 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) :: tmp
    t_0 = x * (y * y)
    if ((y * (x * y)) <= 1d-8) then
        tmp = 1.0d0 / (1.0d0 - t_0)
    else
        tmp = x * (x * (0.16666666666666666d0 * (y * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= 1e-8) {
		tmp = 1.0 / (1.0 - t_0);
	} else {
		tmp = x * (x * (0.16666666666666666 * (y * t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y * y)
	tmp = 0
	if (y * (x * y)) <= 1e-8:
		tmp = 1.0 / (1.0 - t_0)
	else:
		tmp = x * (x * (0.16666666666666666 * (y * t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e-8)
		tmp = Float64(1.0 / Float64(1.0 - t_0));
	else
		tmp = Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(y * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y * y);
	tmp = 0.0;
	if ((y * (x * y)) <= 1e-8)
		tmp = 1.0 / (1.0 - t_0);
	else
		tmp = x * (x * (0.16666666666666666 * (y * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e-8], N[(1.0 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.16666666666666666 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.4%

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

   if 1e-8 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 58.3%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e-8)
   (/ 1.0 (- 1.0 (* x (* y y))))
   (fma x (fma x (* 0.5 (* y y)) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-8) {
		tmp = 1.0 / (1.0 - (x * (y * y)));
	} else {
		tmp = fma(x, fma(x, (0.5 * (y * y)), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e-8)
		tmp = Float64(1.0 / Float64(1.0 - Float64(x * Float64(y * y))));
	else
		tmp = fma(x, fma(x, Float64(0.5 * Float64(y * y)), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e-8], N[(1.0 / N[(1.0 - N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.4%

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

   if 1e-8 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-8}:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(y \cdot y\right), y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 64.7%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 19.0

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

   7. Applied rewrites 19.0%

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

4. Final simplification 53.8%

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

Alternative 14: 53.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 64.7%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 19.0

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

   7. Applied rewrites 19.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 19.0%

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

4. Final simplification 53.8%

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

Alternative 15: 66.2% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 67.5

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

5. Applied rewrites 67.5%

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

Alternative 16: 50.6% 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 50.0%

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

Reproduce

herbie shell --seed 2024222 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))