Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 15.4s

Alternatives: 22

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y} \cdot t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+123}:\\ \;\;\;\;\cosh x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (* x x)))))
   (if (<= x 3.5e-5)
     (* (/ (sin y) y) t_0)
     (if (<= x 7.2e+123)
       (*
        (cosh x)
        (+
         1.0
         (*
          (* y y)
          (+
           -0.16666666666666666
           (*
            (* y y)
            (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984)))))))
       (* (sin y) (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 3.5e-5) {
		tmp = (sin(y) / y) * t_0;
	} else if (x <= 7.2e+123) {
		tmp = cosh(x) * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = sin(y) * (t_0 / 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 = 1.0d0 + (0.5d0 * (x * x))
    if (x <= 3.5d-5) then
        tmp = (sin(y) / y) * t_0
    else if (x <= 7.2d+123) then
        tmp = cosh(x) * (1.0d0 + ((y * y) * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0)))))))
    else
        tmp = sin(y) * (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 3.5e-5) {
		tmp = (Math.sin(y) / y) * t_0;
	} else if (x <= 7.2e+123) {
		tmp = Math.cosh(x) * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = Math.sin(y) * (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.5 * (x * x))
	tmp = 0
	if x <= 3.5e-5:
		tmp = (math.sin(y) / y) * t_0
	elif x <= 7.2e+123:
		tmp = math.cosh(x) * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))))
	else:
		tmp = math.sin(y) * (t_0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	tmp = 0.0
	if (x <= 3.5e-5)
		tmp = Float64(Float64(sin(y) / y) * t_0);
	elseif (x <= 7.2e+123)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984)))))));
	else
		tmp = Float64(sin(y) * Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.5 * (x * x));
	tmp = 0.0;
	if (x <= 3.5e-5)
		tmp = (sin(y) / y) * t_0;
	elseif (x <= 7.2e+123)
		tmp = cosh(x) * (1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	else
		tmp = sin(y) * (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5e-5], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 7.2e+123], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4999999999999997e-5

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. sin-lowering-sin.f64 85.4%

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

   5. Simplified 85.4%

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

   if 3.4999999999999997e-5 < x < 7.19999999999999996e123

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 74.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 74.1%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   if 7.19999999999999996e123 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. sin-lowering-sin.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+123}:\\ \;\;\;\;\cosh x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 0.0155:\\ \;\;\;\;\frac{\sin y}{y} \cdot t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+123}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (* x x)))))
   (if (<= x 0.0155)
     (* (/ (sin y) y) t_0)
     (if (<= x 7.2e+123)
       (* (cosh x) (+ 1.0 (* y (* y -0.16666666666666666))))
       (* (sin y) (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 0.0155) {
		tmp = (sin(y) / y) * t_0;
	} else if (x <= 7.2e+123) {
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = sin(y) * (t_0 / 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 = 1.0d0 + (0.5d0 * (x * x))
    if (x <= 0.0155d0) then
        tmp = (sin(y) / y) * t_0
    else if (x <= 7.2d+123) then
        tmp = cosh(x) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = sin(y) * (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 0.0155) {
		tmp = (Math.sin(y) / y) * t_0;
	} else if (x <= 7.2e+123) {
		tmp = Math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = Math.sin(y) * (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.5 * (x * x))
	tmp = 0
	if x <= 0.0155:
		tmp = (math.sin(y) / y) * t_0
	elif x <= 7.2e+123:
		tmp = math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = math.sin(y) * (t_0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	tmp = 0.0
	if (x <= 0.0155)
		tmp = Float64(Float64(sin(y) / y) * t_0);
	elseif (x <= 7.2e+123)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(sin(y) * Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.5 * (x * x));
	tmp = 0.0;
	if (x <= 0.0155)
		tmp = (sin(y) / y) * t_0;
	elseif (x <= 7.2e+123)
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = sin(y) * (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0155], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 7.2e+123], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0155

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. sin-lowering-sin.f64 85.4%

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

   5. Simplified 85.4%

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

   if 0.0155 < x < 7.19999999999999996e123

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 74.1%

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

   5. Simplified 74.1%

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

   if 7.19999999999999996e123 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. sin-lowering-sin.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0155:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+123}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 0.0134:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) (+ 1.0 (* 0.5 (* x x))))))
   (if (<= x 0.0134)
     t_0
     (if (<= x 1.35e+154)
       (* (cosh x) (+ 1.0 (* y (* y -0.16666666666666666))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * (1.0 + (0.5 * (x * x)));
	double tmp;
	if (x <= 0.0134) {
		tmp = t_0;
	} else if (x <= 1.35e+154) {
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(y) / y) * (1.0d0 + (0.5d0 * (x * x)))
    if (x <= 0.0134d0) then
        tmp = t_0
    else if (x <= 1.35d+154) then
        tmp = cosh(x) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(y) / y) * (1.0 + (0.5 * (x * x)));
	double tmp;
	if (x <= 0.0134) {
		tmp = t_0;
	} else if (x <= 1.35e+154) {
		tmp = Math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(y) / y) * (1.0 + (0.5 * (x * x)))
	tmp = 0
	if x <= 0.0134:
		tmp = t_0
	elif x <= 1.35e+154:
		tmp = math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(0.5 * Float64(x * x))))
	tmp = 0.0
	if (x <= 0.0134)
		tmp = t_0;
	elseif (x <= 1.35e+154)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(y) / y) * (1.0 + (0.5 * (x * x)));
	tmp = 0.0;
	if (x <= 0.0134)
		tmp = t_0;
	elseif (x <= 1.35e+154)
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0134], t$95$0, If[LessEqual[x, 1.35e+154], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0134000000000000005 or 1.35000000000000003e154 < x

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

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

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. sin-lowering-sin.f64 87.0%

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

   5. Simplified 87.0%

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

   if 0.0134000000000000005 < x < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0134:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00058:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+238}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00058)
   (/ (sin y) y)
   (if (<= x 3.3e+238)
     (cosh x)
     (*
      (* (* x x) (* x x))
      (+ 0.041666666666666664 (* (* y y) -0.006944444444444444))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00058) {
		tmp = sin(y) / y;
	} else if (x <= 3.3e+238) {
		tmp = cosh(x);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00058d0) then
        tmp = sin(y) / y
    else if (x <= 3.3d+238) then
        tmp = cosh(x)
    else
        tmp = ((x * x) * (x * x)) * (0.041666666666666664d0 + ((y * y) * (-0.006944444444444444d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00058) {
		tmp = Math.sin(y) / y;
	} else if (x <= 3.3e+238) {
		tmp = Math.cosh(x);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00058:
		tmp = math.sin(y) / y
	elif x <= 3.3e+238:
		tmp = math.cosh(x)
	else:
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00058)
		tmp = Float64(sin(y) / y);
	elseif (x <= 3.3e+238)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.006944444444444444)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00058)
		tmp = sin(y) / y;
	elseif (x <= 3.3e+238)
		tmp = cosh(x);
	else
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00058], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 3.3e+238], N[Cosh[x], $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.8e-4

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \]
   4. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 69.0%

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

   5. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \]

   if 5.8e-4 < x < 3.3000000000000001e238

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 77.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 77.8%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\cosh x} \]

   if 3.3000000000000001e238 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot 1 + \color{blue}{\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} + \color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       20. metadata-eval 90.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{144}\right)\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+242}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+242)
   (cosh x)
   (*
    (* (* x x) (* x x))
    (+ 0.041666666666666664 (* (* y y) -0.006944444444444444)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+242) {
		tmp = cosh(x);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2d+242) then
        tmp = cosh(x)
    else
        tmp = ((x * x) * (x * x)) * (0.041666666666666664d0 + ((y * y) * (-0.006944444444444444d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e+242) {
		tmp = Math.cosh(x);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e+242:
		tmp = math.cosh(x)
	else:
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+242)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.006944444444444444)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2e+242)
		tmp = cosh(x);
	else
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+242], N[Cosh[x], $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e242

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 66.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 66.1%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\cosh x} \]

   if 2.0000000000000001e242 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot 1 + \color{blue}{\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} + \color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       20. metadata-eval 90.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{144}\right)\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\\ t_1 := \left(x \cdot x\right) \cdot 0.041666666666666664\\ t_2 := x \cdot \left(0.5 + t\_1\right)\\ t_3 := x \cdot t\_2\\ \mathbf{if}\;x \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{1 + t\_2 \cdot \left(\left(\left(x \cdot x\right) \cdot t\_2\right) \cdot t\_3\right)}{1 + t\_3 \cdot \left(x \cdot \left(x \cdot 0.5\right) + -1\right)}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - t\_0 \cdot \left(\left(x \cdot t\_0\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{0.5 - \left(x \cdot x\right) \cdot t\_0}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+241}:\\ \;\;\;\;x \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))
        (t_1 (* (* x x) 0.041666666666666664))
        (t_2 (* x (+ 0.5 t_1)))
        (t_3 (* x t_2)))
   (if (<= x 2e+48)
     (/
      (+ 1.0 (* t_2 (* (* (* x x) t_2) t_3)))
      (+ 1.0 (* t_3 (+ (* x (* x 0.5)) -1.0))))
     (if (<= x 4e+77)
       (*
        (+ 1.0 (* y (* y -0.16666666666666666)))
        (+
         1.0
         (/
          (* (* x x) (- 0.25 (* t_0 (* (* x t_0) (* x (* x x))))))
          (- 0.5 (* (* x x) t_0)))))
       (if (<= x 5.8e+241)
         (* x (* x t_1))
         (*
          (* (* x x) (* x x))
          (+ 0.041666666666666664 (* (* y y) -0.006944444444444444))))))))

C

double code(double x, double y) {
	double t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double t_1 = (x * x) * 0.041666666666666664;
	double t_2 = x * (0.5 + t_1);
	double t_3 = x * t_2;
	double tmp;
	if (x <= 2e+48) {
		tmp = (1.0 + (t_2 * (((x * x) * t_2) * t_3))) / (1.0 + (t_3 * ((x * (x * 0.5)) + -1.0)));
	} else if (x <= 4e+77) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	} else if (x <= 5.8e+241) {
		tmp = x * (x * t_1);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))
    t_1 = (x * x) * 0.041666666666666664d0
    t_2 = x * (0.5d0 + t_1)
    t_3 = x * t_2
    if (x <= 2d+48) then
        tmp = (1.0d0 + (t_2 * (((x * x) * t_2) * t_3))) / (1.0d0 + (t_3 * ((x * (x * 0.5d0)) + (-1.0d0))))
    else if (x <= 4d+77) then
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + (((x * x) * (0.25d0 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5d0 - ((x * x) * t_0))))
    else if (x <= 5.8d+241) then
        tmp = x * (x * t_1)
    else
        tmp = ((x * x) * (x * x)) * (0.041666666666666664d0 + ((y * y) * (-0.006944444444444444d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double t_1 = (x * x) * 0.041666666666666664;
	double t_2 = x * (0.5 + t_1);
	double t_3 = x * t_2;
	double tmp;
	if (x <= 2e+48) {
		tmp = (1.0 + (t_2 * (((x * x) * t_2) * t_3))) / (1.0 + (t_3 * ((x * (x * 0.5)) + -1.0)));
	} else if (x <= 4e+77) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	} else if (x <= 5.8e+241) {
		tmp = x * (x * t_1);
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889))
	t_1 = (x * x) * 0.041666666666666664
	t_2 = x * (0.5 + t_1)
	t_3 = x * t_2
	tmp = 0
	if x <= 2e+48:
		tmp = (1.0 + (t_2 * (((x * x) * t_2) * t_3))) / (1.0 + (t_3 * ((x * (x * 0.5)) + -1.0)))
	elif x <= 4e+77:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))))
	elif x <= 5.8e+241:
		tmp = x * (x * t_1)
	else:
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))
	t_1 = Float64(Float64(x * x) * 0.041666666666666664)
	t_2 = Float64(x * Float64(0.5 + t_1))
	t_3 = Float64(x * t_2)
	tmp = 0.0
	if (x <= 2e+48)
		tmp = Float64(Float64(1.0 + Float64(t_2 * Float64(Float64(Float64(x * x) * t_2) * t_3))) / Float64(1.0 + Float64(t_3 * Float64(Float64(x * Float64(x * 0.5)) + -1.0))));
	elseif (x <= 4e+77)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.25 - Float64(t_0 * Float64(Float64(x * t_0) * Float64(x * Float64(x * x)))))) / Float64(0.5 - Float64(Float64(x * x) * t_0)))));
	elseif (x <= 5.8e+241)
		tmp = Float64(x * Float64(x * t_1));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.006944444444444444)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	t_1 = (x * x) * 0.041666666666666664;
	t_2 = x * (0.5 + t_1);
	t_3 = x * t_2;
	tmp = 0.0;
	if (x <= 2e+48)
		tmp = (1.0 + (t_2 * (((x * x) * t_2) * t_3))) / (1.0 + (t_3 * ((x * (x * 0.5)) + -1.0)));
	elseif (x <= 4e+77)
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	elseif (x <= 5.8e+241)
		tmp = x * (x * t_1);
	else
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$2), $MachinePrecision]}, If[LessEqual[x, 2e+48], N[(N[(1.0 + N[(t$95$2 * N[(N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$3 * N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e+77], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.25 - N[(t$95$0 * N[(N[(x * t$95$0), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+241], N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.00000000000000009e48

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 63.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 55.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \frac{{1}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right) - 1 \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({1}^{3} + {\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)}^{3}\right), \color{blue}{\left(1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right) - 1 \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right)}\right) \]

   10. Applied egg-rr 40.8%

       \[\leadsto \color{blue}{\frac{1 + \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)}{1 + \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) + -1\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right), -1\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 44.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, x\right)\right), -1\right)\right)\right)\right) \]

   13. Simplified 44.0%

       \[\leadsto \frac{1 + \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)}{1 + \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(0.5 \cdot x\right)} + -1\right)} \]

   if 2.00000000000000009e48 < x < 3.99999999999999993e77

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \left(\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   10. Applied egg-rr 80.0%

       \[\leadsto \left(1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot x\right)}{0.5 - \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)}}\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   if 3.99999999999999993e77 < x < 5.8000000000000004e241

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 86.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 86.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 86.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right) \cdot x \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x \]

       8. unpow3 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       19. *-lowering-*.f64 86.7%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 86.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 5.8000000000000004e241 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot 1 + \color{blue}{\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} + \color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       20. metadata-eval 90.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{144}\right)\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{1 + \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)}{1 + \left(x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \left(x \cdot \left(x \cdot 0.5\right) + -1\right)}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{0.5 - \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+241}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - t\_0 \cdot \left(\left(x \cdot t\_0\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{0.5 - \left(x \cdot x\right) \cdot t\_0}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+240}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* x (* x 0.001388888888888889)))))
   (if (<= x 5e+77)
     (*
      (+ 1.0 (* y (* y -0.16666666666666666)))
      (+
       1.0
       (/
        (* (* x x) (- 0.25 (* t_0 (* (* x t_0) (* x (* x x))))))
        (- 0.5 (* (* x x) t_0)))))
     (if (<= x 5e+240)
       (* x (* x (* (* x x) 0.041666666666666664)))
       (*
        (* (* x x) (* x x))
        (+ 0.041666666666666664 (* (* y y) -0.006944444444444444)))))))

C

double code(double x, double y) {
	double t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double tmp;
	if (x <= 5e+77) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	} else if (x <= 5e+240) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	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 = 0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))
    if (x <= 5d+77) then
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + (((x * x) * (0.25d0 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5d0 - ((x * x) * t_0))))
    else if (x <= 5d+240) then
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    else
        tmp = ((x * x) * (x * x)) * (0.041666666666666664d0 + ((y * y) * (-0.006944444444444444d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	double tmp;
	if (x <= 5e+77) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	} else if (x <= 5e+240) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889))
	tmp = 0
	if x <= 5e+77:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))))
	elif x <= 5e+240:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	else:
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))
	tmp = 0.0
	if (x <= 5e+77)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.25 - Float64(t_0 * Float64(Float64(x * t_0) * Float64(x * Float64(x * x)))))) / Float64(0.5 - Float64(Float64(x * x) * t_0)))));
	elseif (x <= 5e+240)
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.006944444444444444)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.041666666666666664 + (x * (x * 0.001388888888888889));
	tmp = 0.0;
	if (x <= 5e+77)
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + (((x * x) * (0.25 - (t_0 * ((x * t_0) * (x * (x * x)))))) / (0.5 - ((x * x) * t_0))));
	elseif (x <= 5e+240)
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	else
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+77], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.25 - N[(t$95$0 * N[(N[(x * t$95$0), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+240], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.00000000000000004e77

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right), \left(\frac{1}{2} - x \cdot \left(x \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   10. Applied egg-rr 45.1%

       \[\leadsto \left(1 + \color{blue}{\frac{\left(0.25 - \left(\left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot x\right)}{0.5 - \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)}}\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   if 5.00000000000000004e77 < x < 5.0000000000000003e240

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 86.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 86.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 86.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right) \cdot x \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x \]

       8. unpow3 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       19. *-lowering-*.f64 86.7%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 86.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 5.0000000000000003e240 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot \frac{1}{24}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot 1 + \color{blue}{\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} + \color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{24}} \cdot \frac{-1}{6}\right)\right)\right)\right) \]

       20. metadata-eval 90.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{144}\right)\right)\right) \]

   11. Simplified 90.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{0.5 - \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+240}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ t_1 := \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* y (* y -0.16666666666666666)))
          (+
           1.0
           (*
            (* x x)
            (+
             0.5
             (*
              x
              (*
               x
               (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))))
        (t_1
         (*
          (+ 1.0 (* 0.5 (* x x)))
          (+
           1.0
           (*
            y
            (*
             y
             (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))))
   (if (<= y 7.5e+39)
     t_0
     (if (<= y 2.15e+123) t_1 (if (<= y 8.5e+297) t_0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	double t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0;
	} else if (y <= 2.15e+123) {
		tmp = t_1;
	} else if (y <= 8.5e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    t_1 = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    if (y <= 7.5d+39) then
        tmp = t_0
    else if (y <= 2.15d+123) then
        tmp = t_1
    else if (y <= 8.5d+297) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	double t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0;
	} else if (y <= 2.15e+123) {
		tmp = t_1;
	} else if (y <= 8.5e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	tmp = 0
	if y <= 7.5e+39:
		tmp = t_0
	elif y <= 2.15e+123:
		tmp = t_1
	elif y <= 8.5e+297:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))))
	t_1 = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))
	tmp = 0.0
	if (y <= 7.5e+39)
		tmp = t_0;
	elseif (y <= 2.15e+123)
		tmp = t_1;
	elseif (y <= 8.5e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	tmp = 0.0;
	if (y <= 7.5e+39)
		tmp = t_0;
	elseif (y <= 2.15e+123)
		tmp = t_1;
	elseif (y <= 8.5e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+39], t$95$0, If[LessEqual[y, 2.15e+123], t$95$1, If[LessEqual[y, 8.5e+297], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000005e39 or 2.14999999999999993e123 < y < 8.4999999999999994e297

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 72.3%

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

   5. Simplified 72.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   8. Simplified 68.6%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   if 7.5000000000000005e39 < y < 2.14999999999999993e123 or 8.4999999999999994e297 < y

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right) + \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \color{blue}{\left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ t_1 := \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* y (* y -0.16666666666666666)))
          (+
           1.0
           (* (* x x) (+ 0.5 (* x (* x (* (* x x) 0.001388888888888889))))))))
        (t_1
         (*
          (+ 1.0 (* 0.5 (* x x)))
          (+
           1.0
           (*
            y
            (*
             y
             (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))))
   (if (<= y 7.5e+39)
     t_0
     (if (<= y 2.15e+123) t_1 (if (<= y 8.5e+297) t_0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
	double t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0;
	} else if (y <= 2.15e+123) {
		tmp = t_1;
	} else if (y <= 8.5e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * ((x * x) * 0.001388888888888889d0))))))
    t_1 = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    if (y <= 7.5d+39) then
        tmp = t_0
    else if (y <= 2.15d+123) then
        tmp = t_1
    else if (y <= 8.5d+297) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
	double t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0;
	} else if (y <= 2.15e+123) {
		tmp = t_1;
	} else if (y <= 8.5e+297) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))))
	t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	tmp = 0
	if y <= 7.5e+39:
		tmp = t_0
	elif y <= 2.15e+123:
		tmp = t_1
	elif y <= 8.5e+297:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(Float64(x * x) * 0.001388888888888889)))))))
	t_1 = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))
	tmp = 0.0
	if (y <= 7.5e+39)
		tmp = t_0;
	elseif (y <= 2.15e+123)
		tmp = t_1;
	elseif (y <= 8.5e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + (x * (x * ((x * x) * 0.001388888888888889))))));
	t_1 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	tmp = 0.0;
	if (y <= 7.5e+39)
		tmp = t_0;
	elseif (y <= 2.15e+123)
		tmp = t_1;
	elseif (y <= 8.5e+297)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+39], t$95$0, If[LessEqual[y, 2.15e+123], t$95$1, If[LessEqual[y, 8.5e+297], t$95$0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000005e39 or 2.14999999999999993e123 < y < 8.4999999999999994e297

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 72.3%

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

   5. Simplified 72.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   8. Simplified 68.6%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{3}\right)}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 68.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   11. Simplified 68.3%

       \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   if 7.5000000000000005e39 < y < 2.14999999999999993e123 or 8.4999999999999994e297 < y

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right) + \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \color{blue}{\left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ t_1 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{t\_1}{y}}{\frac{1}{y \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* y y) -0.16666666666666666)))
        (t_1 (+ 1.0 (* 0.5 (* x x))))
        (t_2
         (*
          t_1
          (+
           1.0
           (*
            y
            (*
             y
             (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))))
   (if (<= y 7.5e+39)
     (* t_0 (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664))))))
     (if (<= y 2.15e+123)
       t_2
       (if (<= y 8.5e+297) (/ (/ t_1 y) (/ 1.0 (* y t_0))) t_2)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * -0.16666666666666666);
	double t_1 = 1.0 + (0.5 * (x * x));
	double t_2 = t_1 * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0 * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	} else if (y <= 2.15e+123) {
		tmp = t_2;
	} else if (y <= 8.5e+297) {
		tmp = (t_1 / y) / (1.0 / (y * t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    t_1 = 1.0d0 + (0.5d0 * (x * x))
    t_2 = t_1 * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    if (y <= 7.5d+39) then
        tmp = t_0 * (1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0)))))
    else if (y <= 2.15d+123) then
        tmp = t_2
    else if (y <= 8.5d+297) then
        tmp = (t_1 / y) / (1.0d0 / (y * t_0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * -0.16666666666666666);
	double t_1 = 1.0 + (0.5 * (x * x));
	double t_2 = t_1 * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = t_0 * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	} else if (y <= 2.15e+123) {
		tmp = t_2;
	} else if (y <= 8.5e+297) {
		tmp = (t_1 / y) / (1.0 / (y * t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * y) * -0.16666666666666666)
	t_1 = 1.0 + (0.5 * (x * x))
	t_2 = t_1 * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	tmp = 0
	if y <= 7.5e+39:
		tmp = t_0 * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))))
	elif y <= 2.15e+123:
		tmp = t_2
	elif y <= 8.5e+297:
		tmp = (t_1 / y) / (1.0 / (y * t_0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666))
	t_1 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	t_2 = Float64(t_1 * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))
	tmp = 0.0
	if (y <= 7.5e+39)
		tmp = Float64(t_0 * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664))))));
	elseif (y <= 2.15e+123)
		tmp = t_2;
	elseif (y <= 8.5e+297)
		tmp = Float64(Float64(t_1 / y) / Float64(1.0 / Float64(y * t_0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * y) * -0.16666666666666666);
	t_1 = 1.0 + (0.5 * (x * x));
	t_2 = t_1 * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	tmp = 0.0;
	if (y <= 7.5e+39)
		tmp = t_0 * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	elseif (y <= 2.15e+123)
		tmp = t_2;
	elseif (y <= 8.5e+297)
		tmp = (t_1 / y) / (1.0 / (y * t_0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+39], N[(t$95$0 * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+123], t$95$2, If[LessEqual[y, 8.5e+297], N[(N[(t$95$1 / y), $MachinePrecision] / N[(1.0 / N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5000000000000005e39

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   if 7.5000000000000005e39 < y < 2.14999999999999993e123 or 8.4999999999999994e297 < y

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right) + \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \color{blue}{\left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   if 2.14999999999999993e123 < y < 8.4999999999999994e297

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 31.7%

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

   10. Simplified 31.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}}{\frac{1}{y \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+39}:\\ \;\;\;\;\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (+ 1.0 (* 0.5 (* x x)))
          (+
           1.0
           (*
            y
            (*
             y
             (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))))
   (if (<= y 7.5e+39)
     (*
      (+ 1.0 (* (* y y) -0.16666666666666666))
      (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664))))))
     (if (<= y 2.15e+123)
       t_0
       (if (<= y 8.5e+297)
         (+
          1.0
          (*
           y
           (*
            y
            (+
             -0.16666666666666666
             (*
              (* y y)
              (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984)))))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	} else if (y <= 2.15e+123) {
		tmp = t_0;
	} else if (y <= 8.5e+297) {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    if (y <= 7.5d+39) then
        tmp = (1.0d0 + ((y * y) * (-0.16666666666666666d0))) * (1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0)))))
    else if (y <= 2.15d+123) then
        tmp = t_0
    else if (y <= 8.5d+297) then
        tmp = 1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0)))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	double tmp;
	if (y <= 7.5e+39) {
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	} else if (y <= 2.15e+123) {
		tmp = t_0;
	} else if (y <= 8.5e+297) {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	tmp = 0
	if y <= 7.5e+39:
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))))
	elif y <= 2.15e+123:
		tmp = t_0
	elif y <= 8.5e+297:
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))
	tmp = 0.0
	if (y <= 7.5e+39)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664))))));
	elseif (y <= 2.15e+123)
		tmp = t_0;
	elseif (y <= 8.5e+297)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	tmp = 0.0;
	if (y <= 7.5e+39)
		tmp = (1.0 + ((y * y) * -0.16666666666666666)) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	elseif (y <= 2.15e+123)
		tmp = t_0;
	elseif (y <= 8.5e+297)
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+39], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+123], t$95$0, If[LessEqual[y, 8.5e+297], N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5000000000000005e39

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   if 7.5000000000000005e39 < y < 2.14999999999999993e123 or 8.4999999999999994e297 < y

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 67.6%

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

   7. Simplified 67.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right) + \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \color{blue}{\left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

   10. Simplified 47.1%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   if 2.14999999999999993e123 < y < 8.4999999999999994e297

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 31.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 31.7%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 31.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 31.7%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+297}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e+123)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (if (<= y 8.5e+297)
     (+
      1.0
      (*
       y
       (*
        y
        (+
         -0.16666666666666666
         (*
          (* y y)
          (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984)))))))
     (*
      (+ 1.0 (* 0.5 (* x x)))
      (+
       1.0
       (*
        y
        (* y (+ -0.16666666666666666 (* (* y y) 0.008333333333333333)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else if (y <= 8.5e+297) {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d+123) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))))
    else if (y <= 8.5d+297) then
        tmp = 1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0)))))))
    else
        tmp = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else if (y <= 8.5e+297) {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e+123:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))))
	elif y <= 8.5e+297:
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))))
	else:
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e+123)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))));
	elseif (y <= 8.5e+297)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984)))))));
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e+123)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	elseif (y <= 8.5e+297)
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	else
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e+123], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+297], N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.14999999999999993e123

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 74.4%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.6%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 2.14999999999999993e123 < y < 8.4999999999999994e297

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 31.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 31.7%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 31.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 31.7%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   if 8.4999999999999994e297 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. div-inv N/A

         \[\leadsto \cosh x \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{\sin y}}} \]

      3. associate-/r* N/A

         \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cosh x \cdot \frac{1}{y}}{\color{blue}{\frac{1}{\sin y}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{y} \cdot \cosh x}{\frac{\color{blue}{1}}{\sin y}} \]

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

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{y}}{\frac{1}{\sin y}}} \]

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(\frac{1}{2} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(1 + \frac{1}{2} \cdot {x}^{2}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right) + \left(\color{blue}{1} + \frac{1}{2} \cdot {x}^{2}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) \cdot {y}^{2} + \color{blue}{\left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right) + \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot {y}^{2}} + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \]

   10. Simplified 38.5%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e+123)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (+
    1.0
    (*
     y
     (*
      y
      (+
       -0.16666666666666666
       (*
        (* y y)
        (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984)))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d+123) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))))
    else
        tmp = 1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e+123:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))))
	else:
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e+123)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))));
	else
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e+123)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	else
		tmp = 1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e+123], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.14999999999999993e123

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 74.4%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.6%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 2.14999999999999993e123 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 24.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 24.5%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 24.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 24.5%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e+123)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
   (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d+123) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e+123:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e+123)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e+123)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e+123], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.14999999999999993e123

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 74.4%

         \[\leadsto \mathsf{cosh.f64}\left(x\right) \]

   7. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\cosh x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 67.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 67.6%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \]

   if 2.14999999999999993e123 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 24.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 21.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 21.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \frac{-1}{12}\right)\right)\right)\right) \]

       14. metadata-eval N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. metadata-eval 22.1%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 22.1%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{+69}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+237}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e+69)
   (+ 1.0 (* (* y y) -0.16666666666666666))
   (if (<= x 9.5e+237)
     (* x (* x (* (* x x) 0.041666666666666664)))
     (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.8e+69) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else if (x <= 9.5e+237) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.8d+69) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else if (x <= 9.5d+237) then
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.8e+69) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else if (x <= 9.5e+237) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.8e+69:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	elif x <= 9.5e+237:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.8e+69)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	elseif (x <= 9.5e+237)
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.8e+69)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	elseif (x <= 9.5e+237)
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.8e+69], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+237], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.80000000000000028e69

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 43.4%

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

   8. Simplified 43.4%

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

   if 3.80000000000000028e69 < x < 9.50000000000000061e237

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 84.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 81.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 81.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right) \cdot x \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x \]

       8. unpow3 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       19. *-lowering-*.f64 81.5%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 81.5%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 9.50000000000000061e237 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 90.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \frac{-1}{12}\right)\right)\right)\right) \]

       14. metadata-eval N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. metadata-eval 90.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 90.9%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 55.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e+123)
   (+ 1.0 (* x (* x (+ 0.5 (* x (* x 0.041666666666666664))))))
   (* x (* x (+ 0.5 (* (* y y) -0.08333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * 0.041666666666666664)))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d+123) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + (x * (x * 0.041666666666666664d0)))))
    else
        tmp = x * (x * (0.5d0 + ((y * y) * (-0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e+123) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * 0.041666666666666664)))));
	} else {
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.15e+123:
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * 0.041666666666666664)))))
	else:
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.15e+123)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e+123)
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * 0.041666666666666664)))));
	else
		tmp = x * (x * (0.5 + ((y * y) * -0.08333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.15e+123], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.14999999999999993e123

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot \frac{1}{24}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{24}\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 65.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{24}\right), x\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 65.7%

       \[\leadsto 1 + x \cdot \left(x \cdot \left(0.5 + \color{blue}{\left(x \cdot 0.041666666666666664\right) \cdot x}\right)\right) \]

   if 2.14999999999999993e123 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 24.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(\frac{-1}{6} \cdot {y}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{{x}^{2}} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {\color{blue}{x}}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + {x}^{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot 1 + \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

   8. Simplified 21.9%

      \[\leadsto \color{blue}{\left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

   11. Simplified 21.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. distribute-rgt-in N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

       12. associate-*l* N/A

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

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({y}^{2} \cdot \frac{-1}{12}\right)\right)\right)\right) \]

       14. metadata-eval N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. metadata-eval 22.1%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{12}\right)\right)\right)\right) \]

   14. Simplified 22.1%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+123}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+69}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e+69)
   (+ 1.0 (* (* y y) -0.16666666666666666))
   (* x (* x (* (* x x) 0.041666666666666664)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e+69) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6d+69) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6e+69) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e+69:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e+69)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e+69)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e+69], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999967e69

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 43.4%

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

   8. Simplified 43.4%

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

   if 5.99999999999999967e69 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 83.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 81.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right) \cdot x \]

       7. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x \]

       8. unpow3 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{3}\right) \cdot x \]

       9. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       19. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 39.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+123}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e+123) (+ 1.0 (* (* y y) -0.16666666666666666)) (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+123) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.2d+123) then
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+123) {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e+123:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e+123)
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e+123)
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e+123], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.19999999999999996e123

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 65.2%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right) \]

      5. *-lowering-*.f64 41.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right) \]

   8. Simplified 41.3%

      \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666} \]

   if 7.19999999999999996e123 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 86.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 73.7%

       \[\leadsto 1 + x \cdot \left(x \cdot \color{blue}{0.5}\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 73.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   14. Simplified 73.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 8.8e+15) 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8.8d+15) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.8e+15:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.8e+15)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.8e+15)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.8e+15], 1.0, N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8e15

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 63.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 40.2%

      \[\leadsto \color{blue}{1} \]

   if 8.8e15 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 79.6%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 65.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{2}}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 41.2%

       \[\leadsto 1 + x \cdot \left(x \cdot \color{blue}{0.5}\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 41.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   14. Simplified 41.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 45.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.5d0 * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 66.8%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   4. *-lowering-*.f64 49.4%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

8. Simplified 49.4%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]
9. Add Preprocessing

Alternative 22: 26.6% accurate, 205.0× 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 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 66.8%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 32.3%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (cosh(x) * sin(y)) / y
end function

Java

public static double code(double x, double y) {
	return (Math.cosh(x) * Math.sin(y)) / y;
}

Python

def code(x, y):
	return (math.cosh(x) * math.sin(y)) / y

Julia

function code(x, y)
	return Float64(Float64(cosh(x) * sin(y)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (cosh(x) * sin(y)) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Reproduce

herbie shell --seed 2024191 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))