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

Percentage Accurate: 100.0% → 100.0%

Time: 36.0s

Alternatives: 16

Speedup: 1.0×

• 24.1% 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: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := t\_0 \cdot \left(1 + t\_0 \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;y \leq 5.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{t\_1 \cdot t\_1 + -1}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y)))
        (t_1 (* t_0 (+ 1.0 (* t_0 (+ 0.5 (* t_0 0.16666666666666666)))))))
   (if (<= y 5.4e-79) (/ (+ (* t_1 t_1) -1.0) (+ t_1 -1.0)) (exp (* x y)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	double tmp;
	if (y <= 5.4e-79) {
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y * (x * y)
    t_1 = t_0 * (1.0d0 + (t_0 * (0.5d0 + (t_0 * 0.16666666666666666d0))))
    if (y <= 5.4d-79) then
        tmp = ((t_1 * t_1) + (-1.0d0)) / (t_1 + (-1.0d0))
    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 t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	double tmp;
	if (y <= 5.4e-79) {
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	} else {
		tmp = Math.exp((x * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))))
	tmp = 0
	if y <= 5.4e-79:
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0)
	else:
		tmp = math.exp((x * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(t_0 * Float64(1.0 + Float64(t_0 * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))
	tmp = 0.0
	if (y <= 5.4e-79)
		tmp = Float64(Float64(Float64(t_1 * t_1) + -1.0) / Float64(t_1 + -1.0));
	else
		tmp = exp(Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	tmp = 0.0;
	if (y <= 5.4e-79)
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	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]}, Block[{t$95$1 = N[(t$95$0 * N[(1.0 + N[(t$95$0 * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.4e-79], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4000000000000004e-79

   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 72.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 63.8%

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

   if 5.4000000000000004e-79 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 87.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right) + -1}{\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := t\_0 \cdot \left(1 + t\_0 \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{t\_1 \cdot t\_1 + -1}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y)))
        (t_1 (* t_0 (+ 1.0 (* t_0 (+ 0.5 (* t_0 0.16666666666666666)))))))
   (if (<= y 7.2e-79) (/ (+ (* t_1 t_1) -1.0) (+ t_1 -1.0)) (exp x))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	double tmp;
	if (y <= 7.2e-79) {
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (x * y)
    t_1 = t_0 * (1.0d0 + (t_0 * (0.5d0 + (t_0 * 0.16666666666666666d0))))
    if (y <= 7.2d-79) then
        tmp = ((t_1 * t_1) + (-1.0d0)) / (t_1 + (-1.0d0))
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	double tmp;
	if (y <= 7.2e-79) {
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))))
	tmp = 0
	if y <= 7.2e-79:
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(t_0 * Float64(1.0 + Float64(t_0 * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))
	tmp = 0.0
	if (y <= 7.2e-79)
		tmp = Float64(Float64(Float64(t_1 * t_1) + -1.0) / Float64(t_1 + -1.0));
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	t_1 = t_0 * (1.0 + (t_0 * (0.5 + (t_0 * 0.16666666666666666))));
	tmp = 0.0;
	if (y <= 7.2e-79)
		tmp = ((t_1 * t_1) + -1.0) / (t_1 + -1.0);
	else
		tmp = exp(x);
	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[(t$95$0 * N[(1.0 + N[(t$95$0 * N[(0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-79], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], N[Exp[x], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000005e-79

   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 72.6%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 63.8%

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

   if 7.2000000000000005e-79 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 69.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right) \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right) + -1}{\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(y \cdot \left(x \cdot y\right) + 1\right) + x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  (+ (* y (* x y)) 1.0)
  (*
   x
   (*
    (* y y)
    (* (* x (* y y)) (+ 0.5 (* y (* x (* y 0.16666666666666666)))))))))

C

double code(double x, double y) {
	return ((y * (x * y)) + 1.0) + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((y * (x * y)) + 1.0d0) + (x * ((y * y) * ((x * (y * y)) * (0.5d0 + (y * (x * (y * 0.16666666666666666d0)))))))
end function

Java

public static double code(double x, double y) {
	return ((y * (x * y)) + 1.0) + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
}

Python

def code(x, y):
	return ((y * (x * y)) + 1.0) + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))))

Julia

function code(x, y)
	return Float64(Float64(Float64(y * Float64(x * y)) + 1.0) + Float64(x * Float64(Float64(y * y) * Float64(Float64(x * Float64(y * y)) * Float64(0.5 + Float64(y * Float64(x * Float64(y * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((y * (x * y)) + 1.0) + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
end

Wolfram

code[x_, y_] := N[(N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(y * N[(x * N[(y * 0.16666666666666666), $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 69.5%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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 69.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

9. Applied egg-rr 69.9%

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

10. Final simplification 69.9%

    \[\leadsto \left(y \cdot \left(x \cdot y\right) + 1\right) + x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 71.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \left(t\_0 + 1\right) + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (+
    (+ t_0 1.0)
    (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	return (t_0 + 1.0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = y * (x * y)
    code = (t_0 + 1.0d0) + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	return (t_0 + 1.0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
}

Python

def code(x, y):
	t_0 = y * (x * y)
	return (t_0 + 1.0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	return Float64(Float64(t_0 + 1.0) + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = y * (x * y);
	tmp = (t_0 + 1.0) + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 + 1.0), $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. 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 69.5%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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 69.9%

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

8. Final simplification 69.9%

   \[\leadsto \left(y \cdot \left(x \cdot y\right) + 1\right) + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 6: 61.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+135}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-133)
   1.0
   (if (<= y 7.8e+135)
     (+ 1.0 (* (* y y) (* y (* y (* 0.5 (* x x))))))
     (* y (* x (* 0.5 (* x (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-133) {
		tmp = 1.0;
	} else if (y <= 7.8e+135) {
		tmp = 1.0 + ((y * y) * (y * (y * (0.5 * (x * x)))));
	} else {
		tmp = y * (x * (0.5 * (x * (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 <= 1.4d-133) then
        tmp = 1.0d0
    else if (y <= 7.8d+135) then
        tmp = 1.0d0 + ((y * y) * (y * (y * (0.5d0 * (x * x)))))
    else
        tmp = y * (x * (0.5d0 * (x * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-133) {
		tmp = 1.0;
	} else if (y <= 7.8e+135) {
		tmp = 1.0 + ((y * y) * (y * (y * (0.5 * (x * x)))));
	} else {
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e-133:
		tmp = 1.0
	elif y <= 7.8e+135:
		tmp = 1.0 + ((y * y) * (y * (y * (0.5 * (x * x)))))
	else:
		tmp = y * (x * (0.5 * (x * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-133)
		tmp = 1.0;
	elseif (y <= 7.8e+135)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(0.5 * Float64(x * x))))));
	else
		tmp = Float64(y * Float64(x * Float64(0.5 * Float64(x * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-133)
		tmp = 1.0;
	elseif (y <= 7.8e+135)
		tmp = 1.0 + ((y * y) * (y * (y * (0.5 * (x * x)))));
	else
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e-133], 1.0, If[LessEqual[y, 7.8e+135], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 * N[(x * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3999999999999999e-133

   1. Initial program 100.0%

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

   4. Applied egg-rr 60.0%

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

   if 1.3999999999999999e-133 < y < 7.80000000000000064e135

   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 63.8%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 69.4%

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

   8. Simplified 69.4%

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

   if 7.80000000000000064e135 < 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 59.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 59.0%

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

   7. Applied egg-rr 59.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. associate-*r* N/A

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

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

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

   10. Simplified 59.0%

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

Alternative 7: 67.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-79}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{elif}\;y \leq 10^{+103}:\\ \;\;\;\;1 + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-79)
   (+ (* y (* x y)) 1.0)
   (if (<= y 1e+103)
     (+ 1.0 (* x (* (* x x) (* (* y y) (* y 0.16666666666666666)))))
     (* y (* x (* 0.5 (* x (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-79) {
		tmp = (y * (x * y)) + 1.0;
	} else if (y <= 1e+103) {
		tmp = 1.0 + (x * ((x * x) * ((y * y) * (y * 0.16666666666666666))));
	} else {
		tmp = y * (x * (0.5 * (x * (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 <= 6.5d-79) then
        tmp = (y * (x * y)) + 1.0d0
    else if (y <= 1d+103) then
        tmp = 1.0d0 + (x * ((x * x) * ((y * y) * (y * 0.16666666666666666d0))))
    else
        tmp = y * (x * (0.5d0 * (x * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-79) {
		tmp = (y * (x * y)) + 1.0;
	} else if (y <= 1e+103) {
		tmp = 1.0 + (x * ((x * x) * ((y * y) * (y * 0.16666666666666666))));
	} else {
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e-79:
		tmp = (y * (x * y)) + 1.0
	elif y <= 1e+103:
		tmp = 1.0 + (x * ((x * x) * ((y * y) * (y * 0.16666666666666666))))
	else:
		tmp = y * (x * (0.5 * (x * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-79)
		tmp = Float64(Float64(y * Float64(x * y)) + 1.0);
	elseif (y <= 1e+103)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * x) * Float64(Float64(y * y) * Float64(y * 0.16666666666666666)))));
	else
		tmp = Float64(y * Float64(x * Float64(0.5 * Float64(x * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e-79)
		tmp = (y * (x * y)) + 1.0;
	elseif (y <= 1e+103)
		tmp = 1.0 + (x * ((x * x) * ((y * y) * (y * 0.16666666666666666))));
	else
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e-79], N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 1e+103], N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 * N[(x * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.5000000000000003e-79

   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 69.4%

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

   5. Simplified 69.4%

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

   if 6.5000000000000003e-79 < y < 1e103

   1. Initial program 100.0%

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 60.2%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 60.2%

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

   10. Simplified 60.2%

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

   if 1e103 < 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 57.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 57.1%

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

   7. Applied egg-rr 57.1%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. associate-*r* N/A

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

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

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

   10. Simplified 60.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-79}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{elif}\;y \leq 10^{+103}:\\ \;\;\;\;1 + x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ 1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (*
   x
   (*
    (* y y)
    (* (* x (* y y)) (+ 0.5 (* y (* x (* y 0.16666666666666666)))))))))

C

double code(double x, double y) {
	return 1.0 + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (x * ((y * y) * ((x * (y * y)) * (0.5d0 + (y * (x * (y * 0.16666666666666666d0)))))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
}

Python

def code(x, y):
	return 1.0 + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(x * Float64(Float64(y * y) * Float64(Float64(x * Float64(y * y)) * Float64(0.5 + Float64(y * Float64(x * Float64(y * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (x * ((y * y) * ((x * (y * y)) * (0.5 + (y * (x * (y * 0.16666666666666666)))))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(y * N[(x * N[(y * 0.16666666666666666), $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 69.5%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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 69.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

9. Applied egg-rr 69.9%

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

10. Taylor expanded in y around 0

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

12. Simplified 69.5%

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

13. Final simplification 69.5%

    \[\leadsto 1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 9: 71.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ 1 + y \cdot \left(t\_0 \cdot \left(x \cdot \left(y \cdot \left(0.5 + t\_0 \cdot 0.16666666666666666\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (+ 1.0 (* y (* t_0 (* x (* y (+ 0.5 (* t_0 0.16666666666666666)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	return 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = y * (x * y)
    code = 1.0d0 + (y * (t_0 * (x * (y * (0.5d0 + (t_0 * 0.16666666666666666d0))))))
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	return 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
}

Python

def code(x, y):
	t_0 = y * (x * y)
	return 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))))

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	return Float64(1.0 + Float64(y * Float64(t_0 * Float64(x * Float64(y * Float64(0.5 + Float64(t_0 * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = y * (x * y);
	tmp = 1.0 + (y * (t_0 * (x * (y * (0.5 + (t_0 * 0.16666666666666666))))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, 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]]

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 69.5%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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 69.9%

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

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 69.5%

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

11. Final simplification 69.5%

    \[\leadsto 1 + y \cdot \left(\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(x \cdot \left(y \cdot \left(0.5 + \left(y \cdot \left(x \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 10: 65.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.7e+104)
   (+ (* y (* x y)) 1.0)
   (* y (* x (* 0.5 (* x (* y (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.7e+104) {
		tmp = (y * (x * y)) + 1.0;
	} else {
		tmp = y * (x * (0.5 * (x * (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 <= 4.7d+104) then
        tmp = (y * (x * y)) + 1.0d0
    else
        tmp = y * (x * (0.5d0 * (x * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.7e+104) {
		tmp = (y * (x * y)) + 1.0;
	} else {
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.7e+104:
		tmp = (y * (x * y)) + 1.0
	else:
		tmp = y * (x * (0.5 * (x * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.7e+104)
		tmp = Float64(Float64(y * Float64(x * y)) + 1.0);
	else
		tmp = Float64(y * Float64(x * Float64(0.5 * Float64(x * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.7e+104)
		tmp = (y * (x * y)) + 1.0;
	else
		tmp = y * (x * (0.5 * (x * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.7e+104], N[(N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(x * N[(0.5 * N[(x * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.70000000000000017e104

   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 67.3%

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

   5. Simplified 67.3%

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

   if 4.70000000000000017e104 < 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 57.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 57.1%

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

   7. Applied egg-rr 57.1%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. unpow3 N/A

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. associate-*r* N/A

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

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

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

   10. Simplified 60.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(1 + 0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* (* y y) (* x (+ 1.0 (* 0.5 (* x (* y y))))))))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * (x * (1.0 + (0.5 * (x * (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 * (1.0d0 + (0.5d0 * (x * (y * y))))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * (x * (1.0 + (0.5 * (x * (y * y))))));
}

Python

def code(x, y):
	return 1.0 + ((y * y) * (x * (1.0 + (0.5 * (x * (y * y))))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * Float64(x * Float64(1.0 + Float64(0.5 * Float64(x * Float64(y * y)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * (x * (1.0 + (0.5 * (x * (y * y))))));
end

Wolfram

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

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
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 68.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 68.6%

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

7. Applied egg-rr 68.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-lowering-*.f64 69.0%

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

9. Applied egg-rr 69.0%

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

10. Final simplification 69.0%

    \[\leadsto 1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(1 + 0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 12: 69.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(x \cdot \left(y \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* (* y (* x y)) (* y (* x (* y 0.5))))))

C

double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (y * (x * (y * 0.5))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * (x * y)) * (y * (x * (y * 0.5d0))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * (x * y)) * (y * (x * (y * 0.5))));
}

Python

def code(x, y):
	return 1.0 + ((y * (x * y)) * (y * (x * (y * 0.5))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * Float64(x * y)) * Float64(y * Float64(x * Float64(y * 0.5)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * (x * y)) * (y * (x * (y * 0.5))));
end

Wolfram

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

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
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 68.3%

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

6. Taylor expanded in y around inf

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 68.0%

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

8. Simplified 68.0%

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

Alternative 13: 58.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+91}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.45e+91) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45e+91) {
		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 <= 2.45d+91) 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 <= 2.45e+91) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45e+91:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45e+91)
		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 <= 2.45e+91)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45e+91], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45000000000000015e91

   1. Initial program 100.0%

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

   4. Applied egg-rr 59.5%

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

   if 2.45000000000000015e91 < 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 43.0%

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

   5. Simplified 43.0%

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

   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 48.5%

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

   8. Simplified 48.5%

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

Alternative 14: 54.1% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+135}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 7.5e+135) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e+135) {
		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 <= 7.5d+135) 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 <= 7.5e+135) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.5e+135:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e+135)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.5e+135)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e+135], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999947e135

   1. Initial program 100.0%

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

   4. Applied egg-rr 57.5%

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

   if 7.49999999999999947e135 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 83.9%

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

   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

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

      6. associate-*l* N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. associate-+r+ N/A

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

      12. distribute-rgt-in N/A

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

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

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

      14. distribute-rgt-in N/A

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

      15. associate-*l* N/A

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

      18. *-commutative N/A

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

   7. Simplified 47.2%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. associate-*l* N/A

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

      18. distribute-rgt-in N/A

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

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

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

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

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

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

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

      24. *-commutative N/A

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

      25. *-lowering-*.f64 47.3%

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

   10. Simplified 47.3%

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

   11. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 27.8%

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

   13. Simplified 27.8%

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

Alternative 15: 63.8% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-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 63.8%

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

5. Simplified 63.8%

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

6. Final simplification 63.8%

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

Alternative 16: 51.6% 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 52.4%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))