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

Percentage Accurate: 100.0% → 100.0%

Time: 13.8s

Alternatives: 15

Speedup: 1.0×

• 25.4% 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: 93.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.02:\\ \;\;\;\;\frac{1}{\left(1 + \left(x \cdot y\right) \cdot \left(y \cdot t\_0\right)\right) \cdot \left(1 - t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) 0.02)
     (/ 1.0 (* (+ 1.0 (* (* x y) (* y t_0))) (- 1.0 t_0)))
     (+
      1.0
      (*
       (* y y)
       (* x (* x (* x (* y (* (* y (* y y)) 0.16666666666666666))))))))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= 0.02) {
		tmp = 1.0 / ((1.0 + ((x * y) * (y * t_0))) * (1.0 - t_0));
	} else {
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y * y)
    if ((y * (x * y)) <= 0.02d0) then
        tmp = 1.0d0 / ((1.0d0 + ((x * y) * (y * t_0))) * (1.0d0 - t_0))
    else
        tmp = 1.0d0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666d0))))))
    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)) <= 0.02) {
		tmp = 1.0 / ((1.0 + ((x * y) * (y * t_0))) * (1.0 - t_0));
	} else {
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y * y)
	tmp = 0
	if (y * (x * y)) <= 0.02:
		tmp = 1.0 / ((1.0 + ((x * y) * (y * t_0))) * (1.0 - t_0))
	else:
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.02)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(x * y) * Float64(y * t_0))) * Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(x * Float64(x * Float64(x * Float64(y * Float64(Float64(y * Float64(y * y)) * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y * y);
	tmp = 0.0;
	if ((y * (x * y)) <= 0.02)
		tmp = 1.0 / ((1.0 + ((x * y) * (y * t_0))) * (1.0 - t_0));
	else
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	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], 0.02], N[(1.0 / N[(N[(1.0 + N[(N[(x * y), $MachinePrecision] * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * N[(x * N[(y * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.0200000000000000004

   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 62.8%

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

   5. Simplified 62.8%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. frac-times N/A

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

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

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 94.5%

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

   if 0.0200000000000000004 < (*.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 79.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*r/ N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

      20. distribute-lft-out N/A

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

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

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

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

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

      23. *-commutative N/A

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

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

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

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

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

      26. unpow2 N/A

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

      27. *-lowering-*.f64 72.3%

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

   8. Simplified 72.3%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

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

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

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 87.4%

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

   11. Simplified 87.4%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       14. *-lowering-*.f64 88.9%

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

   13. Applied egg-rr 88.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.02:\\ \;\;\;\;\frac{1}{\left(1 + \left(x \cdot y\right) \cdot \left(y \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \left(1 - x \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+45}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2e+45)
   (+ 1.0 (* (* x (* y y)) (+ 1.0 (* x (* y (* y 0.5))))))
   (+
    1.0
    (*
     (* y y)
     (* x (* x (* x (* y (* (* y (* y y)) 0.16666666666666666)))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+45) {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 2d+45) then
        tmp = 1.0d0 + ((x * (y * y)) * (1.0d0 + (x * (y * (y * 0.5d0)))))
    else
        tmp = 1.0d0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+45) {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 2e+45:
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))))
	else:
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e+45)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(y * y)) * Float64(1.0 + Float64(x * Float64(y * Float64(y * 0.5))))));
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(x * Float64(x * Float64(x * Float64(y * Float64(Float64(y * Float64(y * y)) * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 2e+45)
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	else
		tmp = 1.0 + ((y * y) * (x * (x * (x * (y * ((y * (y * y)) * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e+45], N[(1.0 + N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(x * N[(x * N[(x * N[(y * N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.9999999999999999e45

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 62.0%

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

   if 1.9999999999999999e45 < (*.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 83.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*r/ N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

      20. distribute-lft-out N/A

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

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

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

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

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

      23. *-commutative N/A

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

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

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

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

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

      26. unpow2 N/A

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

      27. *-lowering-*.f64 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

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

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

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 91.8%

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

   11. Simplified 91.8%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. associate-*l* N/A

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

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

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

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

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

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

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

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

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

       14. *-lowering-*.f64 93.4%

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

   13. Applied egg-rr 93.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+45}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+45}:\\ \;\;\;\;1 + t\_0 \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0 \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) 2e+45)
     (+ 1.0 (* t_0 (+ 1.0 (* x (* y (* y 0.5))))))
     (+ 1.0 (* t_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)) <= 2e+45) {
		tmp = 1.0 + (t_0 * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = 1.0 + (t_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)) <= 2d+45) then
        tmp = 1.0d0 + (t_0 * (1.0d0 + (x * (y * (y * 0.5d0)))))
    else
        tmp = 1.0d0 + (t_0 * (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)) <= 2e+45) {
		tmp = 1.0 + (t_0 * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = 1.0 + (t_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)) <= 2e+45:
		tmp = 1.0 + (t_0 * (1.0 + (x * (y * (y * 0.5)))))
	else:
		tmp = 1.0 + (t_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)) <= 2e+45)
		tmp = Float64(1.0 + Float64(t_0 * Float64(1.0 + Float64(x * Float64(y * Float64(y * 0.5))))));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(x * Float64(x * Float64(0.16666666666666666 * Float64(Float64(y * 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)) <= 2e+45)
		tmp = 1.0 + (t_0 * (1.0 + (x * (y * (y * 0.5)))));
	else
		tmp = 1.0 + (t_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], 2e+45], N[(1.0 + N[(t$95$0 * N[(1.0 + N[(x * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(x * N[(x * N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.9999999999999999e45

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 62.0%

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

   if 1.9999999999999999e45 < (*.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 83.7%

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

   6. Taylor expanded in y around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. associate-*r/ N/A

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

      18. *-commutative N/A

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

      19. associate-/l* N/A

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

      20. distribute-lft-out N/A

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

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

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

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

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

      23. *-commutative N/A

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

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

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

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

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

      26. unpow2 N/A

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

      27. *-lowering-*.f64 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

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

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

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

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

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

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

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 91.8%

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

   11. Simplified 91.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+45}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+47}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 2e+47)
   (+ 1.0 (* (* x (* y y)) (+ 1.0 (* x (* y (* y 0.5))))))
   (* x (* (* y (* y (* y y))) (* x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+47) {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	}
	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)) <= 2d+47) then
        tmp = 1.0d0 + ((x * (y * y)) * (1.0d0 + (x * (y * (y * 0.5d0)))))
    else
        tmp = x * ((y * (y * (y * y))) * (x * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 2e+47) {
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 2e+47:
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))))
	else:
		tmp = x * ((y * (y * (y * y))) * (x * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e+47)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(y * y)) * Float64(1.0 + Float64(x * Float64(y * Float64(y * 0.5))))));
	else
		tmp = Float64(x * Float64(Float64(y * Float64(y * Float64(y * y))) * Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 2e+47)
		tmp = 1.0 + ((x * (y * y)) * (1.0 + (x * (y * (y * 0.5)))));
	else
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 2e+47], N[(1.0 + N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2.0000000000000001e47

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 61.7%

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

   if 2.0000000000000001e47 < (*.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

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

   5. Simplified 76.4%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 65.5%

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

   8. Simplified 65.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 68.9%

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

   10. Applied egg-rr 68.9%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

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

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

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

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 88.2%

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

   12. Applied egg-rr 88.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+47}:\\ \;\;\;\;1 + \left(x \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;t\_0 + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 2e+45) (+ t_0 1.0) (* x (* (* y (* y (* y y))) (* x 0.5))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 2e+45) {
		tmp = t_0 + 1.0;
	} else {
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (t_0 <= 2d+45) then
        tmp = t_0 + 1.0d0
    else
        tmp = x * ((y * (y * (y * y))) * (x * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 2e+45) {
		tmp = t_0 + 1.0;
	} else {
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if t_0 <= 2e+45:
		tmp = t_0 + 1.0
	else:
		tmp = x * ((y * (y * (y * y))) * (x * 0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 2e+45)
		tmp = Float64(t_0 + 1.0);
	else
		tmp = Float64(x * Float64(Float64(y * Float64(y * Float64(y * y))) * Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (t_0 <= 2e+45)
		tmp = t_0 + 1.0;
	else
		tmp = x * ((y * (y * (y * y))) * (x * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+45], N[(t$95$0 + 1.0), $MachinePrecision], N[(x * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.9999999999999999e45

   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 61.9%

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

   5. Simplified 61.9%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 62.0%

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

   7. Applied egg-rr 62.0%

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

   if 1.9999999999999999e45 < (*.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

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

   5. Simplified 75.2%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 64.5%

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

   8. Simplified 64.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.7%

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

   10. Applied egg-rr 67.7%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

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

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

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

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 86.8%

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

   12. Applied egg-rr 86.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;t\_0 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 0.02) (+ t_0 1.0) (* (* x 0.5) (* (* y y) (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = (x * 0.5) * ((y * y) * (x * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (t_0 <= 0.02d0) then
        tmp = t_0 + 1.0d0
    else
        tmp = (x * 0.5d0) * ((y * y) * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = (x * 0.5) * ((y * y) * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if t_0 <= 0.02:
		tmp = t_0 + 1.0
	else:
		tmp = (x * 0.5) * ((y * y) * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.02)
		tmp = Float64(t_0 + 1.0);
	else
		tmp = Float64(Float64(x * 0.5) * Float64(Float64(y * y) * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (t_0 <= 0.02)
		tmp = t_0 + 1.0;
	else
		tmp = (x * 0.5) * ((y * y) * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.02], N[(t$95$0 + 1.0), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.0200000000000000004

   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 62.8%

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

   5. Simplified 62.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 62.9%

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

   7. Applied egg-rr 62.9%

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

   if 0.0200000000000000004 < (*.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

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

   5. Simplified 71.6%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-lowering-*.f64 77.9%

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

   10. Applied egg-rr 77.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.02:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;t\_0 + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 0.02) (+ t_0 1.0) (* y (* y (* x (* x (* (* y y) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = y * (y * (x * (x * ((y * y) * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (t_0 <= 0.02d0) then
        tmp = t_0 + 1.0d0
    else
        tmp = y * (y * (x * (x * ((y * y) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = y * (y * (x * (x * ((y * y) * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if t_0 <= 0.02:
		tmp = t_0 + 1.0
	else:
		tmp = y * (y * (x * (x * ((y * y) * 0.5))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.02)
		tmp = Float64(t_0 + 1.0);
	else
		tmp = Float64(y * Float64(y * Float64(x * Float64(x * Float64(Float64(y * y) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (t_0 <= 0.02)
		tmp = t_0 + 1.0;
	else
		tmp = y * (y * (x * (x * ((y * y) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.02], N[(t$95$0 + 1.0), $MachinePrecision], N[(y * N[(y * N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.0200000000000000004

   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 62.8%

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

   5. Simplified 62.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 62.9%

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

   7. Applied egg-rr 62.9%

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

   if 0.0200000000000000004 < (*.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

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

   5. Simplified 71.6%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*l* N/A

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

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

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

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

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

      14. *-lowering-*.f64 71.6%

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

   10. Applied egg-rr 71.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.02:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;t\_0 + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(\left(x \cdot y\right) \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 0.02) (+ t_0 1.0) (* y (* y (* y (* x (* (* x y) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = y * (y * (y * (x * ((x * y) * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (t_0 <= 0.02d0) then
        tmp = t_0 + 1.0d0
    else
        tmp = y * (y * (y * (x * ((x * y) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = t_0 + 1.0;
	} else {
		tmp = y * (y * (y * (x * ((x * y) * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if t_0 <= 0.02:
		tmp = t_0 + 1.0
	else:
		tmp = y * (y * (y * (x * ((x * y) * 0.5))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.02)
		tmp = Float64(t_0 + 1.0);
	else
		tmp = Float64(y * Float64(y * Float64(y * Float64(x * Float64(Float64(x * y) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (t_0 <= 0.02)
		tmp = t_0 + 1.0;
	else
		tmp = y * (y * (y * (x * ((x * y) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.02], N[(t$95$0 + 1.0), $MachinePrecision], N[(y * N[(y * N[(y * N[(x * N[(N[(x * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.0200000000000000004

   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 62.8%

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

   5. Simplified 62.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 62.9%

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

   7. Applied egg-rr 62.9%

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

   if 0.0200000000000000004 < (*.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

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

   5. Simplified 71.6%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 64.4%

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

   10. Applied egg-rr 64.4%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*l* N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

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

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

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

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

       15. *-lowering-*.f64 71.5%

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

   12. Applied egg-rr 71.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.02:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(\left(x \cdot y\right) \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+55}:\\ \;\;\;\;t\_0 + 1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 5e+55) (+ t_0 1.0) (* (* y (* y y)) (* y (* x (* x 0.5)))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 5e+55) {
		tmp = t_0 + 1.0;
	} else {
		tmp = (y * (y * y)) * (y * (x * (x * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if (t_0 <= 5d+55) then
        tmp = t_0 + 1.0d0
    else
        tmp = (y * (y * y)) * (y * (x * (x * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 5e+55) {
		tmp = t_0 + 1.0;
	} else {
		tmp = (y * (y * y)) * (y * (x * (x * 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if t_0 <= 5e+55:
		tmp = t_0 + 1.0
	else:
		tmp = (y * (y * y)) * (y * (x * (x * 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 5e+55)
		tmp = Float64(t_0 + 1.0);
	else
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(y * Float64(x * Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (t_0 <= 5e+55)
		tmp = t_0 + 1.0;
	else
		tmp = (y * (y * y)) * (y * (x * (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+55], N[(t$95$0 + 1.0), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5.00000000000000046e55

   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 61.4%

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

   5. Simplified 61.4%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 61.4%

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

   7. Applied egg-rr 61.4%

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

   if 5.00000000000000046e55 < (*.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

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

   5. Simplified 77.8%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 66.7%

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

   8. Simplified 66.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 70.1%

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

   10. Applied egg-rr 70.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+55}:\\ \;\;\;\;y \cdot \left(x \cdot y\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+160}:\\ \;\;\;\;1 + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e+160)
   (+ 1.0 (* x (* y y)))
   (* (* y y) (* y (* y (* x (* x 0.5)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e+160) {
		tmp = 1.0 + (x * (y * y));
	} else {
		tmp = (y * y) * (y * (y * (x * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 1d+160) then
        tmp = 1.0d0 + (x * (y * y))
    else
        tmp = (y * y) * (y * (y * (x * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e+160) {
		tmp = 1.0 + (x * (y * y));
	} else {
		tmp = (y * y) * (y * (y * (x * (x * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 1e+160:
		tmp = 1.0 + (x * (y * y))
	else:
		tmp = (y * y) * (y * (y * (x * (x * 0.5))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e+160)
		tmp = Float64(1.0 + Float64(x * Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(y * Float64(x * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 1e+160)
		tmp = 1.0 + (x * (y * y));
	else
		tmp = (y * y) * (y * (y * (x * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e+160], N[(1.0 + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.00000000000000001e160

   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 58.5%

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

   5. Simplified 58.5%

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

   if 1.00000000000000001e160 < (*.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

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

   5. Simplified 100.0%

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

   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. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 89.7%

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

   8. Simplified 89.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+160}:\\ \;\;\;\;1 + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(x \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 65.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

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

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

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

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

   11. *-lowering-*.f64 66.4%

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

7. Applied egg-rr 66.4%

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

8. Final simplification 66.4%

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

Alternative 13: 57.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.9e+114) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e+114) {
		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 <= 1.9d+114) 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 <= 1.9e+114) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e+114:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e+114)
		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 <= 1.9e+114)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e+114], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e114

   1. Initial program 100.0%

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

   4. Applied egg-rr 59.2%

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

   if 1.9e114 < 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 39.5%

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

   5. Simplified 39.5%

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

   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 39.5%

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

   8. Simplified 39.5%

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

Alternative 14: 66.5% 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 61.5%

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

5. Simplified 61.5%

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

Alternative 15: 50.7% 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 48.9%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))