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

Percentage Accurate: 100.0% → 100.0%

Time: 36.8s

Alternatives: 14

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: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot \left(y \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\\ t_1 := \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(-1 - t\_0\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1 + x \cdot \left(\left(\left(1 + t\_0\right) \cdot \left(y \cdot y\right)\right) \cdot t\_1\right)}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x y) (* y (+ 0.5 (* y (* x (* y 0.16666666666666666)))))))
        (t_1 (* (* x (* y y)) (- -1.0 t_0))))
   (if (<= y 5.2e-66)
     (/ (+ 1.0 (* x (* (* (+ 1.0 t_0) (* y y)) t_1))) (+ 1.0 t_1))
     (exp (* x y)))))

C

double code(double x, double y) {
	double t_0 = (x * y) * (y * (0.5 + (y * (x * (y * 0.16666666666666666)))));
	double t_1 = (x * (y * y)) * (-1.0 - t_0);
	double tmp;
	if (y <= 5.2e-66) {
		tmp = (1.0 + (x * (((1.0 + t_0) * (y * y)) * t_1))) / (1.0 + t_1);
	} 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 = (x * y) * (y * (0.5d0 + (y * (x * (y * 0.16666666666666666d0)))))
    t_1 = (x * (y * y)) * ((-1.0d0) - t_0)
    if (y <= 5.2d-66) then
        tmp = (1.0d0 + (x * (((1.0d0 + t_0) * (y * y)) * t_1))) / (1.0d0 + t_1)
    else
        tmp = exp((x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * y) * (y * (0.5 + (y * (x * (y * 0.16666666666666666)))));
	double t_1 = (x * (y * y)) * (-1.0 - t_0);
	double tmp;
	if (y <= 5.2e-66) {
		tmp = (1.0 + (x * (((1.0 + t_0) * (y * y)) * t_1))) / (1.0 + t_1);
	} else {
		tmp = Math.exp((x * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) * (y * (0.5 + (y * (x * (y * 0.16666666666666666)))))
	t_1 = (x * (y * y)) * (-1.0 - t_0)
	tmp = 0
	if y <= 5.2e-66:
		tmp = (1.0 + (x * (((1.0 + t_0) * (y * y)) * t_1))) / (1.0 + t_1)
	else:
		tmp = math.exp((x * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * Float64(y * Float64(0.5 + Float64(y * Float64(x * Float64(y * 0.16666666666666666))))))
	t_1 = Float64(Float64(x * Float64(y * y)) * Float64(-1.0 - t_0))
	tmp = 0.0
	if (y <= 5.2e-66)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(Float64(1.0 + t_0) * Float64(y * y)) * t_1))) / Float64(1.0 + t_1));
	else
		tmp = exp(Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) * (y * (0.5 + (y * (x * (y * 0.16666666666666666)))));
	t_1 = (x * (y * y)) * (-1.0 - t_0);
	tmp = 0.0;
	if (y <= 5.2e-66)
		tmp = (1.0 + (x * (((1.0 + t_0) * (y * y)) * t_1))) / (1.0 + t_1);
	else
		tmp = exp((x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * N[(y * N[(0.5 + N[(y * N[(x * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e-66], N[(N[(1.0 + N[(x * N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999998e-66

   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 76.0%

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

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

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

   7. Applied egg-rr 76.0%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

   9. Applied egg-rr 77.0%

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

   11. Applied egg-rr 63.3%

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

   if 5.1999999999999998e-66 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 90.0%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{1 + x \cdot \left(\left(\left(1 + \left(x \cdot y\right) \cdot \left(y \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(-1 - \left(x \cdot y\right) \cdot \left(y \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)\right)}{1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(-1 - \left(x \cdot y\right) \cdot \left(y \cdot \left(0.5 + y \cdot \left(x \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+93}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(x \cdot \left(0.5 + y \cdot \left(\left(x \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+93)
   (exp x)
   (+
    1.0
    (*
     x
     (*
      (* y y)
      (+
       1.0
       (* y (* y (* x (+ 0.5 (* y (* (* x y) 0.16666666666666666))))))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+93) {
		tmp = exp(x);
	} else {
		tmp = 1.0 + (x * ((y * y) * (1.0 + (y * (y * (x * (0.5 + (y * ((x * y) * 0.16666666666666666)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+93)) then
        tmp = exp(x)
    else
        tmp = 1.0d0 + (x * ((y * y) * (1.0d0 + (y * (y * (x * (0.5d0 + (y * ((x * y) * 0.16666666666666666d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+93) {
		tmp = Math.exp(x);
	} else {
		tmp = 1.0 + (x * ((y * y) * (1.0 + (y * (y * (x * (0.5 + (y * ((x * y) * 0.16666666666666666)))))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e+93:
		tmp = math.exp(x)
	else:
		tmp = 1.0 + (x * ((y * y) * (1.0 + (y * (y * (x * (0.5 + (y * ((x * y) * 0.16666666666666666)))))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+93)
		tmp = exp(x);
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(y * y) * Float64(1.0 + Float64(y * Float64(y * Float64(x * Float64(0.5 + Float64(y * Float64(Float64(x * y) * 0.16666666666666666))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+93)
		tmp = exp(x);
	else
		tmp = 1.0 + (x * ((y * y) * (1.0 + (y * (y * (x * (0.5 + (y * ((x * y) * 0.16666666666666666)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e+93], N[Exp[x], $MachinePrecision], N[(1.0 + N[(x * N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(x * N[(0.5 + N[(y * N[(N[(x * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e93

   1. Initial program 100.0%

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

   4. Applied egg-rr 75.2%

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

   if -1.1500000000000001e93 < x

   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 78.9%

      \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
   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. +-lowering-+.f64 N/A

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

   7. Applied egg-rr 78.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

   9. Applied egg-rr 79.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+93}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(x \cdot \left(0.5 + y \cdot \left(\left(x \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+84}:\\ \;\;\;\;1 + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-66)
   (+ 1.0 (* y (* x y)))
   (if (<= y 4.8e+84)
     (+ 1.0 (* x (* (* x x) (* 0.16666666666666666 (* y (* y y))))))
     (if (<= y 1.35e+154)
       (* (* y y) (* y (* x (* 0.16666666666666666 (* x x)))))
       (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 4.8e+84) {
		tmp = 1.0 + (x * ((x * x) * (0.16666666666666666 * (y * (y * y)))));
	} else if (y <= 1.35e+154) {
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	} 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 <= 5.2d-66) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 4.8d+84) then
        tmp = 1.0d0 + (x * ((x * x) * (0.16666666666666666d0 * (y * (y * y)))))
    else if (y <= 1.35d+154) then
        tmp = (y * y) * (y * (x * (0.16666666666666666d0 * (x * x))))
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 4.8e+84) {
		tmp = 1.0 + (x * ((x * x) * (0.16666666666666666 * (y * (y * y)))));
	} else if (y <= 1.35e+154) {
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-66:
		tmp = 1.0 + (y * (x * y))
	elif y <= 4.8e+84:
		tmp = 1.0 + (x * ((x * x) * (0.16666666666666666 * (y * (y * y)))))
	elif y <= 1.35e+154:
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))))
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-66)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 4.8e+84)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * x) * Float64(0.16666666666666666 * Float64(y * Float64(y * y))))));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(x * x)))));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-66)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 4.8e+84)
		tmp = 1.0 + (x * ((x * x) * (0.16666666666666666 * (y * (y * y)))));
	elseif (y <= 1.35e+154)
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-66], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+84], N[(1.0 + N[(x * N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(y * y), $MachinePrecision] * N[(y * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.1999999999999998e-66

   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 71.3%

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

   5. Simplified 71.3%

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

   if 5.1999999999999998e-66 < y < 4.7999999999999999e84

   1. Initial program 100.0%

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

   4. Applied egg-rr 93.8%

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

   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 74.4%

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

   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. *-lowering-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.4%

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

   10. Simplified 74.4%

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

   if 4.7999999999999999e84 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   4. Applied egg-rr 75.8%

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

   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 45.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. cube-mult N/A

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

      22. unpow2 N/A

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

      23. associate-*l* N/A

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

      24. *-commutative N/A

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

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

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

      26. *-commutative N/A

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

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

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

   10. Simplified 69.4%

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

   if 1.35000000000000003e154 < 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 34.1%

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

   5. Simplified 34.1%

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

   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 46.0%

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

   8. Simplified 46.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+84}:\\ \;\;\;\;1 + x \cdot \left(\left(x \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+83}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.8e-66)
   (+ 1.0 (* y (* x y)))
   (if (<= y 3.9e+83)
     (+ 1.0 (* x (* y (* y (* x 0.5)))))
     (if (<= y 1.35e+154)
       (* (* y y) (* y (* x (* 0.16666666666666666 (* x x)))))
       (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.9e+83) {
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	} else if (y <= 1.35e+154) {
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	} 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 <= 5.8d-66) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 3.9d+83) then
        tmp = 1.0d0 + (x * (y * (y * (x * 0.5d0))))
    else if (y <= 1.35d+154) then
        tmp = (y * y) * (y * (x * (0.16666666666666666d0 * (x * x))))
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.9e+83) {
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	} else if (y <= 1.35e+154) {
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.8e-66:
		tmp = 1.0 + (y * (x * y))
	elif y <= 3.9e+83:
		tmp = 1.0 + (x * (y * (y * (x * 0.5))))
	elif y <= 1.35e+154:
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))))
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.8e-66)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 3.9e+83)
		tmp = Float64(1.0 + Float64(x * Float64(y * Float64(y * Float64(x * 0.5)))));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(x * Float64(0.16666666666666666 * Float64(x * x)))));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.8e-66)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 3.9e+83)
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	elseif (y <= 1.35e+154)
		tmp = (y * y) * (y * (x * (0.16666666666666666 * (x * x))));
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.8e-66], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+83], N[(1.0 + N[(x * N[(y * N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(y * y), $MachinePrecision] * N[(y * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.80000000000000023e-66

   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 71.3%

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

   5. Simplified 71.3%

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

   if 5.80000000000000023e-66 < y < 3.9000000000000002e83

   1. Initial program 100.0%

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

   4. Applied egg-rr 93.8%

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

   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. +-lowering-+.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 74.4%

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

   10. Simplified 74.4%

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

   if 3.9000000000000002e83 < y < 1.35000000000000003e154

   1. Initial program 100.0%

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

   4. Applied egg-rr 75.8%

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

   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 45.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. cube-mult N/A

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

      22. unpow2 N/A

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

      23. associate-*l* N/A

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

      24. *-commutative N/A

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

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

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

      26. *-commutative N/A

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

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

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

   10. Simplified 69.4%

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

   if 1.35000000000000003e154 < 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 34.1%

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

   5. Simplified 34.1%

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

   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 46.0%

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

   8. Simplified 46.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+83}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 + (x * ((y * y) * (1.0 + (y * (y * (x * (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) * (1.0d0 + (y * (y * (x * (0.5d0 + (y * ((x * y) * 0.16666666666666666d0)))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(x * N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(x * N[(0.5 + N[(y * N[(N[(x * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $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 70.9%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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. +-lowering-+.f64 N/A

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

7. Applied egg-rr 71.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

9. Applied egg-rr 71.7%

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

10. Final simplification 71.7%

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

Alternative 7: 72.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(x * N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 * N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $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 70.9%

   \[\leadsto \color{blue}{1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(1 + \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)} \]
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. +-lowering-+.f64 N/A

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

7. Applied egg-rr 71.0%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

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

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

9. Applied egg-rr 71.7%

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

10. Taylor expanded in y around inf

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

    1. unpow3 N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

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

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

    7. *-commutative N/A

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

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

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

    9. unpow2 N/A

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

    10. associate-*l* N/A

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

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

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

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

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

    13. unpow2 N/A

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

    14. *-lowering-*.f64 71.7%

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

12. Simplified 71.7%

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

13. Final simplification 71.7%

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

Alternative 8: 65.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-66)
   (+ 1.0 (* y (* x y)))
   (if (<= y 3.7e+133)
     (+ 1.0 (* x (* y (* y (* x 0.5)))))
     (* x (* 0.5 (* x (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.7e+133) {
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	} else {
		tmp = x * (0.5 * (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 <= 5.5d-66) then
        tmp = 1.0d0 + (y * (x * y))
    else if (y <= 3.7d+133) then
        tmp = 1.0d0 + (x * (y * (y * (x * 0.5d0))))
    else
        tmp = x * (0.5d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-66) {
		tmp = 1.0 + (y * (x * y));
	} else if (y <= 3.7e+133) {
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	} else {
		tmp = x * (0.5 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e-66:
		tmp = 1.0 + (y * (x * y))
	elif y <= 3.7e+133:
		tmp = 1.0 + (x * (y * (y * (x * 0.5))))
	else:
		tmp = x * (0.5 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-66)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	elseif (y <= 3.7e+133)
		tmp = Float64(1.0 + Float64(x * Float64(y * Float64(y * Float64(x * 0.5)))));
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-66)
		tmp = 1.0 + (y * (x * y));
	elseif (y <= 3.7e+133)
		tmp = 1.0 + (x * (y * (y * (x * 0.5))));
	else
		tmp = x * (0.5 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e-66], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+133], N[(1.0 + N[(x * N[(y * N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.50000000000000053e-66

   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 71.3%

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

   5. Simplified 71.3%

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

   if 5.50000000000000053e-66 < y < 3.70000000000000023e133

   1. Initial program 100.0%

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

   4. Applied egg-rr 88.6%

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

   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. +-lowering-+.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 61.1%

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

   7. Simplified 61.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 61.1%

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

   10. Simplified 61.1%

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

   if 3.70000000000000023e133 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 91.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 46.8%

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

   10. Simplified 46.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;1 + x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
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 69.1%

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

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

7. Applied egg-rr 69.9%

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

8. Final simplification 69.9%

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

Alternative 10: 69.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

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 + (t_0 * (1.0d0 + (0.5d0 * t_0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(t$95$0 * N[(1.0 + N[(0.5 * t$95$0), $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 69.1%

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

6. Final simplification 69.1%

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

Alternative 11: 64.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+134}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+134) (+ 1.0 (* y (* x y))) (* x (* 0.5 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+134) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (0.5 * (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 <= 1.05d+134) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = x * (0.5d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+134) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (0.5 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+134:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = x * (0.5 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+134)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e+134)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = x * (0.5 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+134], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e134

   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 67.2%

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

   5. Simplified 67.2%

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

   if 1.05e134 < y

   1. Initial program 100.0%

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

   4. Applied egg-rr 91.7%

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

   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. +-lowering-+.f64 N/A

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

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

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 46.8%

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

   10. Simplified 46.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+134}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+161}:\\ \;\;\;\;1 + y \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e+161) (+ 1.0 (* y (* x y))) (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+161) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d+161) then
        tmp = 1.0d0 + (y * (x * y))
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e+161) {
		tmp = 1.0 + (y * (x * y));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e+161:
		tmp = 1.0 + (y * (x * y))
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e+161)
		tmp = Float64(1.0 + Float64(y * Float64(x * y)));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e+161)
		tmp = 1.0 + (y * (x * y));
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e+161], N[(1.0 + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999986e161

   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 66.1%

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

   5. Simplified 66.1%

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

   if 6.79999999999999986e161 < 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 37.6%

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

   5. Simplified 37.6%

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

   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 50.8%

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

   8. Simplified 50.8%

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

Alternative 13: 57.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.2e+76) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+76) {
		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 <= 3.2d+76) 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 <= 3.2e+76) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e+76:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e+76)
		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 <= 3.2e+76)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e+76], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.19999999999999976e76

   1. Initial program 100.0%

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

   4. Applied egg-rr 60.9%

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

   if 3.19999999999999976e76 < 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 34.7%

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

   5. Simplified 34.7%

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

   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 38.4%

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

   8. Simplified 38.4%

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

Alternative 14: 51.3% 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 50.6%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))