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

Percentage Accurate: 100.0% → 100.0%

Time: 18.8s

Alternatives: 11

Speedup: 1.0×

• 24.6% 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: 92.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(0.5 + y \cdot \left(\left(x \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -20000000.0)
   (/
    1.0
    (-
     1.0
     (*
      x
      (+
       (* y y)
       (*
        x
        (*
         (* y y)
         (* y (* y (+ 0.5 (* y (* (* x y) 0.16666666666666666)))))))))))
   (+
    1.0
    (*
     (* x (* y y))
     (+ 1.0 (* x (* x (* 0.16666666666666666 (* y (* y (* y y)))))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 - (x * ((y * y) + (x * ((y * y) * (y * (y * (0.5 + (y * ((x * y) * 0.16666666666666666))))))))));
	} else {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= (-20000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 - (x * ((y * y) + (x * ((y * y) * (y * (y * (0.5d0 + (y * ((x * y) * 0.16666666666666666d0))))))))))
    else
        tmp = 1.0d0 + ((x * (y * y)) * (1.0d0 + (x * (x * (0.16666666666666666d0 * (y * (y * (y * y))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 - (x * ((y * y) + (x * ((y * y) * (y * (y * (0.5 + (y * ((x * y) * 0.16666666666666666))))))))));
	} else {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= -20000000.0:
		tmp = 1.0 / (1.0 - (x * ((y * y) + (x * ((y * y) * (y * (y * (0.5 + (y * ((x * y) * 0.16666666666666666))))))))))
	else:
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -20000000.0)
		tmp = Float64(1.0 / Float64(1.0 - Float64(x * Float64(Float64(y * y) + Float64(x * Float64(Float64(y * y) * Float64(y * Float64(y * Float64(0.5 + Float64(y * Float64(Float64(x * y) * 0.16666666666666666)))))))))));
	else
		tmp = Float64(1.0 + Float64(Float64(x * Float64(y * y)) * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(y * Float64(y * Float64(y * y)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= -20000000.0)
		tmp = 1.0 / (1.0 - (x * ((y * y) + (x * ((y * y) * (y * (y * (0.5 + (y * ((x * y) * 0.16666666666666666))))))))));
	else
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(1.0 / N[(1.0 - N[(x * N[(N[(y * y), $MachinePrecision] + N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(0.5 + N[(y * N[(N[(x * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e7

   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 1.7%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 0.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 89.5%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

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

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

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

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

       5. associate-*l* N/A

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

       6. associate-*l* N/A

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

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

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

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

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

   12. Applied egg-rr 95.5%

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

   if -2e7 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

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

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y + x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(0.5 + y \cdot \left(\left(x \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(-1 - t\_0 \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -20000000.0)
     (/
      1.0
      (+
       1.0
       (*
        x
        (*
         (* y y)
         (- -1.0 (* t_0 (+ 0.5 (* x (* (* y y) 0.16666666666666666)))))))))
     (+
      1.0
      (*
       t_0
       (+ 1.0 (* x (* x (* 0.16666666666666666 (* y (* y (* y y))))))))))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + (x * ((y * y) * (-1.0 - (t_0 * (0.5 + (x * ((y * y) * 0.16666666666666666))))))));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y * y)
    if ((y * (x * y)) <= (-20000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + (x * ((y * y) * ((-1.0d0) - (t_0 * (0.5d0 + (x * ((y * y) * 0.16666666666666666d0))))))))
    else
        tmp = 1.0d0 + (t_0 * (1.0d0 + (x * (x * (0.16666666666666666d0 * (y * (y * (y * y))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + (x * ((y * y) * (-1.0 - (t_0 * (0.5 + (x * ((y * y) * 0.16666666666666666))))))));
	} else {
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y * y)
	tmp = 0
	if (y * (x * y)) <= -20000000.0:
		tmp = 1.0 / (1.0 + (x * ((y * y) * (-1.0 - (t_0 * (0.5 + (x * ((y * y) * 0.16666666666666666))))))))
	else:
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -20000000.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * Float64(Float64(y * y) * Float64(-1.0 - Float64(t_0 * Float64(0.5 + Float64(x * Float64(Float64(y * y) * 0.16666666666666666)))))))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(y * Float64(y * Float64(y * y)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y * y);
	tmp = 0.0;
	if ((y * (x * y)) <= -20000000.0)
		tmp = 1.0 / (1.0 + (x * ((y * y) * (-1.0 - (t_0 * (0.5 + (x * ((y * y) * 0.16666666666666666))))))));
	else
		tmp = 1.0 + (t_0 * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(1.0 / N[(1.0 + N[(x * N[(N[(y * y), $MachinePrecision] * N[(-1.0 - N[(t$95$0 * N[(0.5 + N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e7

   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 1.7%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 0.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 89.5%

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

   if -2e7 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

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

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + x \cdot \left(\left(y \cdot y\right) \cdot \left(-1 - \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 + x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -20000000.0)
   (/ 1.0 (+ 1.0 (* (* y y) (- (* y (* y (* -0.5 (* x x)))) x))))
   (+
    1.0
    (*
     (* x (* y y))
     (+ 1.0 (* x (* x (* 0.16666666666666666 (* y (* y (* y y)))))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	} else {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= (-20000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + ((y * y) * ((y * (y * ((-0.5d0) * (x * x)))) - x)))
    else
        tmp = 1.0d0 + ((x * (y * y)) * (1.0d0 + (x * (x * (0.16666666666666666d0 * (y * (y * (y * y))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	} else {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= -20000000.0:
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)))
	else:
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -20000000.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(y * Float64(y * Float64(-0.5 * Float64(x * x)))) - x))));
	else
		tmp = Float64(1.0 + Float64(Float64(x * Float64(y * y)) * Float64(1.0 + Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(y * Float64(y * Float64(y * y)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= -20000000.0)
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	else
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (x * (0.16666666666666666 * (y * (y * (y * y))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(1.0 / N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * N[(y * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.16666666666666666 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e7

   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 1.7%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 0.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 89.5%

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

   11. Taylor expanded in y around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 80.7%

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

   13. Simplified 80.7%

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

   if -2e7 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. cube-mult N/A

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

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

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

      19. cube-mult N/A

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

      20. unpow2 N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;\frac{1}{1 + x \cdot \left(y \cdot \left(y \cdot \left(-1 - x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.002)
   (/ 1.0 (+ 1.0 (* x (* y (* y (- -1.0 (* x (* (* y y) 0.5))))))))
   (* x (* x (* 0.5 (* y (* y (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.002) {
		tmp = 1.0 / (1.0 + (x * (y * (y * (-1.0 - (x * ((y * y) * 0.5)))))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 0.002d0) then
        tmp = 1.0d0 / (1.0d0 + (x * (y * (y * ((-1.0d0) - (x * ((y * y) * 0.5d0)))))))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.002) {
		tmp = 1.0 / (1.0 + (x * (y * (y * (-1.0 - (x * ((y * y) * 0.5)))))));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 0.002:
		tmp = 1.0 / (1.0 + (x * (y * (y * (-1.0 - (x * ((y * y) * 0.5)))))))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.002)
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * Float64(y * Float64(y * Float64(-1.0 - Float64(x * Float64(Float64(y * y) * 0.5))))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 0.002)
		tmp = 1.0 / (1.0 + (x * (y * (y * (-1.0 - (x * ((y * y) * 0.5)))))));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.002], N[(1.0 / N[(1.0 + N[(x * N[(y * N[(y * N[(-1.0 - N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 69.1%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 68.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in y around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 92.9%

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

   13. Simplified 92.9%

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

   if 2e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 81.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

      4. rgt-mult-inverse N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. associate-/l* N/A

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

      8. unpow2 N/A

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

      9. unpow3 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \frac{{x}^{3}}{x} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot {x}^{3}}{\color{blue}{x}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\frac{1}{2} \cdot {y}^{4}}{x} \cdot \color{blue}{{x}^{3}} \]

      12. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{{y}^{4}}{x}\right) \cdot {\color{blue}{x}}^{3} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. associate-*r/ N/A

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

      19. associate-*l/ N/A

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

      20. associate-/l* N/A

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

      21. unpow2 N/A

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

      22. associate-/l* N/A

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

      23. *-rgt-identity N/A

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

      24. associate-*r/ N/A

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

      25. rgt-mult-inverse N/A

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

   8. Simplified 90.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;\frac{1}{1 + x \cdot \left(y \cdot \left(y \cdot \left(-1 - x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\left(x \cdot y\right) \cdot \left(1 + x \cdot \left(y \cdot \frac{y}{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -20000000.0)
   (/ 1.0 (+ 1.0 (* (* y y) (- (* y (* y (* -0.5 (* x x)))) x))))
   (+ 1.0 (* y (* (* x y) (+ 1.0 (* x (* y (/ y 2.0)))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	} else {
		tmp = 1.0 + (y * ((x * y) * (1.0 + (x * (y * (y / 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= (-20000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + ((y * y) * ((y * (y * ((-0.5d0) * (x * x)))) - x)))
    else
        tmp = 1.0d0 + (y * ((x * y) * (1.0d0 + (x * (y * (y / 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	} else {
		tmp = 1.0 + (y * ((x * y) * (1.0 + (x * (y * (y / 2.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= -20000000.0:
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)))
	else:
		tmp = 1.0 + (y * ((x * y) * (1.0 + (x * (y * (y / 2.0))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -20000000.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(y * Float64(y * Float64(-0.5 * Float64(x * x)))) - x))));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(x * y) * Float64(1.0 + Float64(x * Float64(y * Float64(y / 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= -20000000.0)
		tmp = 1.0 / (1.0 + ((y * y) * ((y * (y * (-0.5 * (x * x)))) - x)));
	else
		tmp = 1.0 + (y * ((x * y) * (1.0 + (x * (y * (y / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(1.0 / N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * N[(y * N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(x * y), $MachinePrecision] * N[(1.0 + N[(x * N[(y * N[(y / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e7

   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 1.7%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 0.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 89.5%

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

   11. Taylor expanded in y around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 80.7%

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

   13. Simplified 80.7%

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

   if -2e7 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 94.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

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

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

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

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

      14. *-lowering-*.f64 95.3%

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

   7. Applied egg-rr 95.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(-0.5 \cdot \left(x \cdot x\right)\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(\left(x \cdot y\right) \cdot \left(1 + x \cdot \left(y \cdot \frac{y}{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.002)
   (/ 1.0 (- 1.0 (* x (* y y))))
   (* x (* x (* 0.5 (* y (* y (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.002) {
		tmp = 1.0 / (1.0 - (x * (y * y)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 0.002d0) then
        tmp = 1.0d0 / (1.0d0 - (x * (y * y)))
    else
        tmp = x * (x * (0.5d0 * (y * (y * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.002) {
		tmp = 1.0 / (1.0 - (x * (y * y)));
	} else {
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 0.002:
		tmp = 1.0 / (1.0 - (x * (y * y)))
	else:
		tmp = x * (x * (0.5 * (y * (y * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.002)
		tmp = Float64(1.0 / Float64(1.0 - Float64(x * Float64(y * y))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 0.002)
		tmp = 1.0 / (1.0 - (x * (y * y)));
	else
		tmp = x * (x * (0.5 * (y * (y * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.002], N[(1.0 / N[(1.0 - N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 69.1%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 68.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 86.4%

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

   13. Simplified 86.4%

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

   if 2e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-in N/A

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

   5. Simplified 81.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-rgt-identity N/A

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

      4. rgt-mult-inverse N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. associate-/l* N/A

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

      8. unpow2 N/A

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

      9. unpow3 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \frac{{x}^{3}}{x} \]

      10. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot {x}^{3}}{\color{blue}{x}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\frac{1}{2} \cdot {y}^{4}}{x} \cdot \color{blue}{{x}^{3}} \]

      12. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{{y}^{4}}{x}\right) \cdot {\color{blue}{x}}^{3} \]

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. associate-*r/ N/A

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

      19. associate-*l/ N/A

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

      20. associate-/l* N/A

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

      21. unpow2 N/A

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

      22. associate-/l* N/A

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

      23. *-rgt-identity N/A

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

      24. associate-*r/ N/A

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

      25. rgt-mult-inverse N/A

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

   8. Simplified 90.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) -20000000.0) (/ 1.0 (- 1.0 t_0)) (+ 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 - t_0);
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y * y)
    if ((y * (x * y)) <= (-20000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 - t_0)
    else
        tmp = 1.0d0 + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= -20000000.0) {
		tmp = 1.0 / (1.0 - t_0);
	} else {
		tmp = 1.0 + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y * y)
	tmp = 0
	if (y * (x * y)) <= -20000000.0:
		tmp = 1.0 / (1.0 - t_0)
	else:
		tmp = 1.0 + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -20000000.0)
		tmp = Float64(1.0 / Float64(1.0 - t_0));
	else
		tmp = Float64(1.0 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y * y);
	tmp = 0.0;
	if ((y * (x * y)) <= -20000000.0)
		tmp = 1.0 / (1.0 - t_0);
	else
		tmp = 1.0 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -20000000.0], N[(1.0 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -2e7

   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 1.7%

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

      1. flip-+ N/A

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

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

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

   7. Applied egg-rr 0.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 89.5%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 58.5%

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

   13. Simplified 58.5%

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

   if -2e7 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 90.5%

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

   5. Simplified 90.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -20000000:\\ \;\;\;\;\frac{1}{1 - x \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.002) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.002) {
		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 * (x * y)) <= 0.002d0) 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 * (x * y)) <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 0.002:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.002)
		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 * (x * y)) <= 0.002)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.002], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e-3

   1. Initial program 100.0%

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

   4. Applied egg-rr 68.7%

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

   if 2e-3 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.2%

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

   5. Simplified 67.2%

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

   6. Taylor expanded in x 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 67.2%

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

   8. Simplified 67.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 68.4%

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

5. Simplified 68.4%

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

Alternative 11: 51.2% 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 55.4%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))