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

Percentage Accurate: 100.0% → 100.0%

Time: 37.5s

Alternatives: 16

Speedup: 1.0×

• 24.7% 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: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;\left(1 + t\_0\right) + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= y 2.6e-57)
     (+
      (+ 1.0 t_0)
      (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))
     (exp (* x y)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (y <= 2.6e-57) {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else {
		tmp = exp((x * 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 (y <= 2.6d-57) then
        tmp = (1.0d0 + t_0) + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
    else
        tmp = exp((x * 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 (y <= 2.6e-57) {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else {
		tmp = Math.exp((x * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if y <= 2.6e-57:
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))
	else:
		tmp = math.exp((x * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (y <= 2.6e-57)
		tmp = Float64(Float64(1.0 + t_0) + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))));
	else
		tmp = exp(Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (y <= 2.6e-57)
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	else
		tmp = exp((x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.6e-57], N[(N[(1.0 + t$95$0), $MachinePrecision] + N[(y * N[(t$95$0 * N[(x * N[(y * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999985e-57

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 76.5%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 77.6%

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

   if 2.59999999999999985e-57 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 85.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;\left(1 + y \cdot \left(x \cdot y\right)\right) + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_0\right) + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= x -3.7e+25)
     (exp x)
     (+
      (+ 1.0 t_0)
      (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666))))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (x <= -3.7e+25) {
		tmp = exp(x);
	} else {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	}
	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 (x <= (-3.7d+25)) then
        tmp = exp(x)
    else
        tmp = (1.0d0 + t_0) + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (x <= -3.7e+25) {
		tmp = Math.exp(x);
	} else {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if x <= -3.7e+25:
		tmp = math.exp(x)
	else:
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (x <= -3.7e+25)
		tmp = exp(x);
	else
		tmp = Float64(Float64(1.0 + t_0) + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (x <= -3.7e+25)
		tmp = exp(x);
	else
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+25], N[Exp[x], $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] + N[(y * N[(t$95$0 * N[(x * N[(y * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6999999999999999e25

   1. Initial program 100.0%

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

   4. Applied egg-rr 68.6%

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

   if -3.6999999999999999e25 < x

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 81.0%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 82.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+25}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(x \cdot y\right)\right) + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\left(1 + t\_0\right) + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot t\_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))) (t_1 (* y (* y (* y y)))))
   (if (<= y 9.5e+81)
     (+
      (+ 1.0 t_0)
      (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))
     (if (<= y 5e+150)
       (* (* y y) (* (* y y) (/ x (* (/ 1.0 (* y y)) t_1))))
       (+ 1.0 (* t_0 (+ 1.0 (* x (* x (* 0.16666666666666666 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = y * (y * (y * y));
	double tmp;
	if (y <= 9.5e+81) {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 5e+150) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * 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 = y * (y * (y * y))
    if (y <= 9.5d+81) then
        tmp = (1.0d0 + t_0) + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
    else if (y <= 5d+150) then
        tmp = (y * y) * ((y * y) * (x / ((1.0d0 / (y * y)) * t_1)))
    else
        tmp = 1.0d0 + (t_0 * (1.0d0 + (x * (x * (0.16666666666666666d0 * 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 = y * (y * (y * y));
	double tmp;
	if (y <= 9.5e+81) {
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 5e+150) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	t_1 = y * (y * (y * y))
	tmp = 0
	if y <= 9.5e+81:
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))
	elif y <= 5e+150:
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)))
	else:
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(y * Float64(y * Float64(y * y)))
	tmp = 0.0
	if (y <= 9.5e+81)
		tmp = Float64(Float64(1.0 + t_0) + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))));
	elseif (y <= 5e+150)
		tmp = Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x / Float64(Float64(1.0 / Float64(y * y)) * t_1))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	t_1 = y * (y * (y * y));
	tmp = 0.0;
	if (y <= 9.5e+81)
		tmp = (1.0 + t_0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	elseif (y <= 5e+150)
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	else
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.5e+81], N[(N[(1.0 + t$95$0), $MachinePrecision] + N[(y * N[(t$95$0 * N[(x * N[(y * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+150], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x / N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.50000000000000083e81

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 74.7%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 75.7%

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

   if 9.50000000000000083e81 < y < 5.00000000000000009e150

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 8.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 3.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 3.2%

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

      1. unpow1 N/A

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

      2. unpow1 N/A

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

      3. pow-prod-down N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. inv-pow N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y \cdot y} \cdot \left(y \cdot y\right)\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y \cdot y}\right), \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right) \]

      18. *-lowering-*.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   10. Applied egg-rr 10.4%

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

       1. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       2. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot {y}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{\left(-1 + 2\right)}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       5. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{-1} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       6. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       7. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{y \cdot y}\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       14. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

   12. Applied egg-rr 68.6%

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

   if 5.00000000000000009e150 < 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. Simplified 47.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left({x}^{2} \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(x \cdot x\right) \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {y}^{4}\right)\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({y}^{4} \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      19. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 50.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\left(1 + y \cdot \left(x \cdot y\right)\right) + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;1 + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+147}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot t\_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))) (t_1 (* y (* y (* y y)))))
   (if (<= y 1.7e+82)
     (+ 1.0 (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))
     (if (<= y 4e+147)
       (* (* y y) (* (* y y) (/ x (* (/ 1.0 (* y y)) t_1))))
       (+ 1.0 (* t_0 (+ 1.0 (* x (* x (* 0.16666666666666666 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = y * (y * (y * y));
	double tmp;
	if (y <= 1.7e+82) {
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 4e+147) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * 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 = y * (y * (y * y))
    if (y <= 1.7d+82) then
        tmp = 1.0d0 + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
    else if (y <= 4d+147) then
        tmp = (y * y) * ((y * y) * (x / ((1.0d0 / (y * y)) * t_1)))
    else
        tmp = 1.0d0 + (t_0 * (1.0d0 + (x * (x * (0.16666666666666666d0 * 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 = y * (y * (y * y));
	double tmp;
	if (y <= 1.7e+82) {
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 4e+147) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	t_1 = y * (y * (y * y))
	tmp = 0
	if y <= 1.7e+82:
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))
	elif y <= 4e+147:
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)))
	else:
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(y * Float64(y * Float64(y * y)))
	tmp = 0.0
	if (y <= 1.7e+82)
		tmp = Float64(1.0 + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))));
	elseif (y <= 4e+147)
		tmp = Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x / Float64(Float64(1.0 / Float64(y * y)) * t_1))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	t_1 = y * (y * (y * y));
	tmp = 0.0;
	if (y <= 1.7e+82)
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	elseif (y <= 4e+147)
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * t_1)));
	else
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.7e+82], N[(1.0 + N[(y * N[(t$95$0 * N[(x * N[(y * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+147], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x / N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.69999999999999997e82

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 74.7%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 75.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 75.5%

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

   if 1.69999999999999997e82 < y < 3.9999999999999999e147

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 8.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 3.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 3.2%

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

      1. unpow1 N/A

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

      2. unpow1 N/A

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

      3. pow-prod-down N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. inv-pow N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y \cdot y} \cdot \left(y \cdot y\right)\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y \cdot y}\right), \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right) \]

      18. *-lowering-*.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   10. Applied egg-rr 10.4%

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

       1. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       2. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot {y}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{\left(-1 + 2\right)}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       5. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{-1} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       6. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       7. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{y \cdot y}\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       14. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

   12. Applied egg-rr 68.6%

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

   if 3.9999999999999999e147 < 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. Simplified 47.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left({x}^{2} \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(\left(x \cdot x\right) \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot \left(x \cdot {y}^{4}\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {y}^{4}\right)\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot {y}^{4}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({y}^{4} \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot \color{blue}{{y}^{4}}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      19. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 50.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 50.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+82}:\\ \;\;\;\;1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+147}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;1 + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(t\_0 \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot t\_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))) (t_1 (* y (* y y))))
   (if (<= y 2.7e+82)
     (+ 1.0 (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))
     (if (<= y 5.4e+152)
       (* (* y y) (* (* y y) (/ x (* (/ 1.0 (* y y)) (* y t_1)))))
       (+ 1.0 (* y (* t_0 (* x (* 0.16666666666666666 (* x t_1))))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = y * (y * y);
	double tmp;
	if (y <= 2.7e+82) {
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 5.4e+152) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_1))));
	} else {
		tmp = 1.0 + (y * (t_0 * (x * (0.16666666666666666 * (x * 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 = y * (y * y)
    if (y <= 2.7d+82) then
        tmp = 1.0d0 + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
    else if (y <= 5.4d+152) then
        tmp = (y * y) * ((y * y) * (x / ((1.0d0 / (y * y)) * (y * t_1))))
    else
        tmp = 1.0d0 + (y * (t_0 * (x * (0.16666666666666666d0 * (x * 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 = y * (y * y);
	double tmp;
	if (y <= 2.7e+82) {
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	} else if (y <= 5.4e+152) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_1))));
	} else {
		tmp = 1.0 + (y * (t_0 * (x * (0.16666666666666666 * (x * t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	t_1 = y * (y * y)
	tmp = 0
	if y <= 2.7e+82:
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))
	elif y <= 5.4e+152:
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_1))))
	else:
		tmp = 1.0 + (y * (t_0 * (x * (0.16666666666666666 * (x * t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (y <= 2.7e+82)
		tmp = Float64(1.0 + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))));
	elseif (y <= 5.4e+152)
		tmp = Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x / Float64(Float64(1.0 / Float64(y * y)) * Float64(y * t_1)))));
	else
		tmp = Float64(1.0 + Float64(y * Float64(t_0 * Float64(x * Float64(0.16666666666666666 * Float64(x * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	t_1 = y * (y * y);
	tmp = 0.0;
	if (y <= 2.7e+82)
		tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
	elseif (y <= 5.4e+152)
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_1))));
	else
		tmp = 1.0 + (y * (t_0 * (x * (0.16666666666666666 * (x * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.7e+82], N[(1.0 + N[(y * N[(t$95$0 * N[(x * N[(y * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e+152], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x / N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(t$95$0 * N[(x * N[(0.16666666666666666 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6999999999999999e82

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 74.7%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 75.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 75.5%

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

   if 2.6999999999999999e82 < y < 5.4000000000000003e152

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 8.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 3.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 3.2%

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

      1. unpow1 N/A

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

      2. unpow1 N/A

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

      3. pow-prod-down N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. inv-pow N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y \cdot y} \cdot \left(y \cdot y\right)\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y \cdot y}\right), \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right) \]

      18. *-lowering-*.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   10. Applied egg-rr 10.4%

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

       1. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       2. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot {y}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{\left(-1 + 2\right)}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       5. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{-1} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       6. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       7. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{y \cdot y}\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       14. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

   12. Applied egg-rr 68.6%

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

   if 5.4000000000000003e152 < 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. Simplified 47.3%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 47.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 47.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{3}\right)\right)}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left({x}^{2} \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot {y}^{3}\right)\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\left(x \cdot {y}^{3}\right) \cdot \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left({y}^{3}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot {y}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 50.8%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

   13. Simplified 50.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ t_1 := 1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot t\_0\right)\right)\right)\right)\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y)))
        (t_1
         (+
          1.0
          (* y (* (* y (* x y)) (* x (* 0.16666666666666666 (* x t_0))))))))
   (if (<= y 2.8e+82)
     t_1
     (if (<= y 5e+150)
       (* (* y y) (* (* y y) (/ x (* (/ 1.0 (* y y)) (* y t_0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double t_1 = 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * t_0)))));
	double tmp;
	if (y <= 2.8e+82) {
		tmp = t_1;
	} else if (y <= 5e+150) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_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 * (y * y)
    t_1 = 1.0d0 + (y * ((y * (x * y)) * (x * (0.16666666666666666d0 * (x * t_0)))))
    if (y <= 2.8d+82) then
        tmp = t_1
    else if (y <= 5d+150) then
        tmp = (y * y) * ((y * y) * (x / ((1.0d0 / (y * y)) * (y * t_0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * y);
	double t_1 = 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * t_0)))));
	double tmp;
	if (y <= 2.8e+82) {
		tmp = t_1;
	} else if (y <= 5e+150) {
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * y)
	t_1 = 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * t_0)))))
	tmp = 0
	if y <= 2.8e+82:
		tmp = t_1
	elif y <= 5e+150:
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	t_1 = Float64(1.0 + Float64(y * Float64(Float64(y * Float64(x * y)) * Float64(x * Float64(0.16666666666666666 * Float64(x * t_0))))))
	tmp = 0.0
	if (y <= 2.8e+82)
		tmp = t_1;
	elseif (y <= 5e+150)
		tmp = Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x / Float64(Float64(1.0 / Float64(y * y)) * Float64(y * t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * y);
	t_1 = 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * t_0)))));
	tmp = 0.0;
	if (y <= 2.8e+82)
		tmp = t_1;
	elseif (y <= 5e+150)
		tmp = (y * y) * ((y * y) * (x / ((1.0 / (y * y)) * (y * t_0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(y * N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.8e+82], t$95$1, If[LessEqual[y, 5e+150], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x / N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e82 or 5.00000000000000009e150 < y

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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. Simplified 71.7%

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

   7. Applied egg-rr 72.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 72.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{3}\right)\right)}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left({x}^{2} \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot {y}^{3}\right)\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\left(x \cdot {y}^{3}\right) \cdot \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left({y}^{3}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot {y}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 72.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

   13. Simplified 72.3%

       \[\leadsto 1 + y \cdot \left(\color{blue}{\left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right) \]

   if 2.8e82 < y < 5.00000000000000009e150

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 8.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 3.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 3.2%

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

      1. unpow1 N/A

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

      2. unpow1 N/A

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

      3. pow-prod-down N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. inv-pow N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y \cdot y} \cdot \left(y \cdot y\right)\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x}{y \cdot y}\right), \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right), \left(y \cdot y\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right)\right) \]

      18. *-lowering-*.f64 10.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   10. Applied egg-rr 10.4%

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

       1. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot y\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       2. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({y}^{1} \cdot {y}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{1}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{\left(-1 + 2\right)}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       5. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left({\left(y \cdot y\right)}^{-1} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       6. inv-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot {\left(y \cdot y\right)}^{2}\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       7. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{1}{y \cdot y}\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

       14. *-lowering-*.f64 68.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right)\right) \]

   12. Applied egg-rr 68.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+82}:\\ \;\;\;\;1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{x}{\frac{1}{y \cdot y} \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-57}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+97}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(1 + x \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.6e-57)
   (+ 1.0 (* y (* x y)))
   (if (<= y 3.3e+97)
     (+ 1.0 (* x (* y (+ 1.0 (* x (* y 0.5))))))
     (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.6e-57) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.3e+97) {
		tmp = 1.0 + (x * (y * (1.0 + (x * (y * 0.5)))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 <= 8.6d-57) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 3.3d+97) then
        tmp = 1.0d0 + (x * (y * (1.0d0 + (x * (y * 0.5d0)))))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.6e-57) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.3e+97) {
		tmp = 1.0 + (x * (y * (1.0 + (x * (y * 0.5)))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.6e-57:
		tmp = 1.0 + (y * (x * y))
	elif y <= 3.3e+97:
		tmp = 1.0 + (x * (y * (1.0 + (x * (y * 0.5)))))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.6e-57)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 3.3e+97)
		tmp = Float64(1.0 + Float64(x * Float64(y * Float64(1.0 + Float64(x * Float64(y * 0.5))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.6e-57)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 3.3e+97)
		tmp = 1.0 + (x * (y * (1.0 + (x * (y * 0.5)))));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.6e-57], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+97], N[(1.0 + N[(x * N[(y * N[(1.0 + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.60000000000000043e-57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 69.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 69.9%

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

   if 8.60000000000000043e-57 < y < 3.3000000000000001e97

   1. Initial program 100.0%

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

   4. Applied egg-rr 96.0%

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

   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. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. associate-+r+ N/A

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

      13. distribute-rgt-in N/A

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

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y \cdot \left(x + \frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)\right)}\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot y + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right) \cdot y}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \color{blue}{\left(\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)\right)} \cdot y\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot x + \frac{1}{2} \cdot \color{blue}{\left(\left({x}^{2} \cdot y\right) \cdot y\right)}\right)\right) \]

   7. Simplified 53.8%

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

   if 3.3000000000000001e97 < 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

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. fma-define N/A

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

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 35.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 42.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 42.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-57}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+97}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(1 + x \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{+96}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + t\_0 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= y 8.5e+96)
     (+ 1.0 (* t_0 (+ 1.0 (* t_0 0.5))))
     (* x (* x (* 0.5 (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (y <= 8.5e+96) {
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 (y <= 8.5d+96) then
        tmp = 1.0d0 + (t_0 * (1.0d0 + (t_0 * 0.5d0)))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (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 (y <= 8.5e+96) {
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if y <= 8.5e+96:
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (y <= 8.5e+96)
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(t_0 * 0.5))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (y <= 8.5e+96)
		tmp = 1.0 + (t_0 * (1.0 + (t_0 * 0.5)));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8.5e+96], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000025e96

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. fma-define N/A

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

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 72.5%

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

   if 8.50000000000000025e96 < 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

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. fma-define N/A

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

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 34.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 41.4%

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

Alternative 10: 70.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (* y (* (* y (* x y)) (* x (* 0.16666666666666666 (* x (* y (* y y)))))))))

C

double code(double x, double y) {
	return 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * (y * (y * y)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (y * ((y * (x * y)) * (x * (0.16666666666666666d0 * (x * (y * (y * y)))))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * (y * (y * y)))))));
}

Python

def code(x, y):
	return 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * (y * (y * y)))))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(y * Float64(Float64(y * Float64(x * y)) * Float64(x * Float64(0.16666666666666666 * Float64(x * Float64(y * Float64(y * y))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (y * ((y * (x * y)) * (x * (0.16666666666666666 * (x * (y * (y * y)))))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(y * N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(x * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

5. Simplified 67.8%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(1 + y \cdot \left(x \cdot y\right)\right), \color{blue}{\left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(x \cdot y\right)\right)\right), \left(\color{blue}{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(x \cdot y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)}\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)}\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(\left(y \cdot \left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \cdot \left(\color{blue}{y} \cdot \left(x \cdot y\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \left(y \cdot \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)\right)}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot y\right) \cdot \left(\frac{1}{2} + x \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \color{blue}{\left(y \cdot \left(x \cdot y\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 68.5%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right), \frac{1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 68.2%

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

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{3}\right)\right)}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left({x}^{2} \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot {y}^{3}\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\left(x \cdot \left(x \cdot {y}^{3}\right)\right) \cdot \frac{1}{6}\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\left(x \cdot {y}^{3}\right) \cdot \frac{1}{6}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot {y}^{3}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left({y}^{3}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    9. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(y \cdot {y}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 68.1%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, y\right)\right)\right)\right)\right) \]

13. Simplified 68.1%

    \[\leadsto 1 + y \cdot \left(\color{blue}{\left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \cdot \left(y \cdot \left(x \cdot y\right)\right)\right) \]

14. Final simplification 68.1%

    \[\leadsto 1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right) \]
15. Add Preprocessing

Alternative 11: 65.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+97}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e+97)
   (+ 1.0 (* y (* x y)))
   (* x (* x (* 0.5 (* y (* y (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.4e+97) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (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 <= 3.4d+97) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e+97) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.4e+97:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.4e+97)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e+97)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.4e+97], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4000000000000001e97

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 65.8%

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

   if 3.4000000000000001e97 < 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

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. fma-define N/A

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

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot {y}^{4}, \frac{1}{2}, 1 + x \cdot {y}^{2}\right) \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. fma-define N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

   5. Simplified 35.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{4}\right)}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

      11. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{3}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 42.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 42.3%

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

Alternative 12: 64.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+116}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e+116) (+ 1.0 (* y (* x y))) (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+116) {
		tmp = 1.0 + (y * (x * y));
	} 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 (y <= 6.8d+116) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+116) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e+116:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e+116)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e+116)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e+116], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000046e116

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 63.5%

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

   if 6.80000000000000046e116 < 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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 24.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 24.0%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 33.9%

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

Alternative 13: 57.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+96}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.6e+96) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.6e+96) {
		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 (y <= 8.6d+96) 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 (y <= 8.6e+96) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.6e+96:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.6e+96)
		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 (y <= 8.6e+96)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.6e+96], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.60000000000000003e96

   1. Initial program 99.9%

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

   4. Applied egg-rr 57.4%

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

   if 8.60000000000000003e96 < 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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

      6. *-lowering-*.f64 20.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 20.9%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 28.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 28.2%

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

Alternative 14: 53.2% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.7e+155) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.7e+155) {
		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 <= 1.7d+155) 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 <= 1.7e+155) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.7e+155:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.7e+155)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.7e+155)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.7e+155], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7e155

   1. Initial program 99.9%

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

   4. Applied egg-rr 52.6%

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

   if 1.7e155 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 72.9%

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

   5. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 11.0%

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

   8. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 11.0%

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

Alternative 15: 66.1% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + (x * (y * y))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot {y}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(x \cdot y\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot y\right)}\right)\right) \]

   6. *-lowering-*.f64 57.8%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

5. Simplified 57.8%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \color{blue}{x}\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \color{blue}{x}\right)\right) \]

   4. *-lowering-*.f64 61.4%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right) \]

7. Applied egg-rr 61.4%

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

8. Final simplification 61.4%

   \[\leadsto 1 + x \cdot \left(y \cdot y\right) \]
9. Add Preprocessing

Alternative 16: 50.6% accurate, 105.0× 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

4. Applied egg-rr 47.8%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))