Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 13.2s

Alternatives: 23

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 94.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\frac{\cos x}{\frac{1}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\cos x}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 42.0)
   (/
    (cos x)
    (/
     1.0
     (+
      1.0
      (*
       y
       (*
        y
        (+
         0.16666666666666666
         (*
          y
          (*
           y
           (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))))
   (if (<= y 3.6e+44)
     (/ (sinh y) y)
     (/
      1.0
      (/
       1.0
       (/
        (cos x)
        (/
         y
         (*
          y
          (+
           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 <= 42.0) {
		tmp = cos(x) / (1.0 / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))));
	} else if (y <= 3.6e+44) {
		tmp = sinh(y) / y;
	} else {
		tmp = 1.0 / (1.0 / (cos(x) / (y / (y * (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 <= 42.0d0) then
        tmp = cos(x) / (1.0d0 / (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))))
    else if (y <= 3.6d+44) then
        tmp = sinh(y) / y
    else
        tmp = 1.0d0 / (1.0d0 / (cos(x) / (y / (y * (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 <= 42.0) {
		tmp = Math.cos(x) / (1.0 / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))));
	} else if (y <= 3.6e+44) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = 1.0 / (1.0 / (Math.cos(x) / (y / (y * (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 <= 42.0:
		tmp = math.cos(x) / (1.0 / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))))
	elif y <= 3.6e+44:
		tmp = math.sinh(y) / y
	else:
		tmp = 1.0 / (1.0 / (math.cos(x) / (y / (y * (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 <= 42.0)
		tmp = Float64(cos(x) / Float64(1.0 / Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))));
	elseif (y <= 3.6e+44)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(cos(x) / Float64(y / Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 42.0)
		tmp = cos(x) / (1.0 / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))));
	elseif (y <= 3.6e+44)
		tmp = sinh(y) / y;
	else
		tmp = 1.0 / (1.0 / (cos(x) / (y / (y * (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, 42.0], N[(N[Cos[x], $MachinePrecision] / N[(1.0 / N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+44], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[Cos[x], $MachinePrecision] / N[(y / N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 42

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \sinh y}{\color{blue}{y}} \]

      2. div-inv N/A

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

      3. sinh-def N/A

         \[\leadsto \left(\cos x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right) \cdot \frac{1}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2} \cdot \frac{\color{blue}{1}}{y} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}{\color{blue}{2}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}}} \]

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

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

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

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

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

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

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

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

      15. div-inv N/A

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

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

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

      17. sinh-undef N/A

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

      18. associate-/r* N/A

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\cos x}{y \cdot \frac{0.5}{\sinh y}}}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. sinh-def N/A

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

      14. clear-num N/A

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

      15. frac-times N/A

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

      16. clear-num N/A

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

      17. sinh-def N/A

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\cos x}{\frac{y}{\sinh y}}}}} \]

   7. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 92.8%

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

   9. Simplified 92.8%

      \[\leadsto \frac{1}{\frac{1}{\frac{\cos x}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}}}}} \]
   10. Step-by-step derivation

       1. remove-double-div N/A

          \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\frac{y}{y}}{\color{blue}{1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)}}\right)\right) \]

       5. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{1}{\color{blue}{1} + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right)\right)\right) \]

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

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

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

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

   11. Applied egg-rr 91.9%

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

   if 42 < y < 3.6e44

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 66.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 3.6e44 < y

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \sinh y}{\color{blue}{y}} \]

      2. div-inv N/A

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

      3. sinh-def N/A

         \[\leadsto \left(\cos x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right) \cdot \frac{1}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2} \cdot \frac{\color{blue}{1}}{y} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}{\color{blue}{2}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}}} \]

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

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

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

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

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

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

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

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

      15. div-inv N/A

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

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

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

      17. sinh-undef N/A

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

      18. associate-/r* N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\cos x}{y \cdot \frac{0.5}{\sinh y}}}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. sinh-def N/A

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

      14. clear-num N/A

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

      15. frac-times N/A

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

      16. clear-num N/A

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

      17. sinh-def N/A

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

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\cos x}{\frac{y}{\sinh y}}}}} \]

   7. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{1}{\frac{1}{\frac{\cos x}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\frac{\cos x}{\frac{1}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\cos x}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\\ \mathbf{if}\;y \leq 42:\\ \;\;\;\;\frac{\cos x}{\frac{1}{1 + y \cdot \left(y \cdot t\_0\right)}}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          0.16666666666666666
          (*
           y
           (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
   (if (<= y 42.0)
     (/ (cos x) (/ 1.0 (+ 1.0 (* y (* y t_0)))))
     (if (<= y 9.8e+51) (/ (sinh y) y) (* (cos x) (+ 1.0 (* (* y y) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 42.0) {
		tmp = cos(x) / (1.0 / (1.0 + (y * (y * t_0))));
	} else if (y <= 9.8e+51) {
		tmp = sinh(y) / y;
	} else {
		tmp = cos(x) * (1.0 + ((y * y) * 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 = 0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))
    if (y <= 42.0d0) then
        tmp = cos(x) / (1.0d0 / (1.0d0 + (y * (y * t_0))))
    else if (y <= 9.8d+51) then
        tmp = sinh(y) / y
    else
        tmp = cos(x) * (1.0d0 + ((y * y) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 42.0) {
		tmp = Math.cos(x) / (1.0 / (1.0 + (y * (y * t_0))));
	} else if (y <= 9.8e+51) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = Math.cos(x) * (1.0 + ((y * y) * t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))
	tmp = 0
	if y <= 42.0:
		tmp = math.cos(x) / (1.0 / (1.0 + (y * (y * t_0))))
	elif y <= 9.8e+51:
		tmp = math.sinh(y) / y
	else:
		tmp = math.cos(x) * (1.0 + ((y * y) * t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))
	tmp = 0.0
	if (y <= 42.0)
		tmp = Float64(cos(x) / Float64(1.0 / Float64(1.0 + Float64(y * Float64(y * t_0)))));
	elseif (y <= 9.8e+51)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	tmp = 0.0;
	if (y <= 42.0)
		tmp = cos(x) / (1.0 / (1.0 + (y * (y * t_0))));
	elseif (y <= 9.8e+51)
		tmp = sinh(y) / y;
	else
		tmp = cos(x) * (1.0 + ((y * y) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 42.0], N[(N[Cos[x], $MachinePrecision] / N[(1.0 / N[(1.0 + N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e+51], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 42

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \sinh y}{\color{blue}{y}} \]

      2. div-inv N/A

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

      3. sinh-def N/A

         \[\leadsto \left(\cos x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right) \cdot \frac{1}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2} \cdot \frac{\color{blue}{1}}{y} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}{\color{blue}{2}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}}} \]

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

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

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

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

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

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

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

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

      15. div-inv N/A

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

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

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

      17. sinh-undef N/A

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

      18. associate-/r* N/A

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\cos x}{y \cdot \frac{0.5}{\sinh y}}}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. sinh-def N/A

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

      14. clear-num N/A

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

      15. frac-times N/A

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

      16. clear-num N/A

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

      17. sinh-def N/A

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\cos x}{\frac{y}{\sinh y}}}}} \]

   7. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 92.8%

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

   9. Simplified 92.8%

      \[\leadsto \frac{1}{\frac{1}{\frac{\cos x}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}}}}} \]
   10. Step-by-step derivation

       1. remove-double-div N/A

          \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\frac{y}{y}}{\color{blue}{1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)}}\right)\right) \]

       5. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{1}{\color{blue}{1} + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + y \cdot \left(y \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right)\right)\right) \]

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

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

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

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

   11. Applied egg-rr 91.9%

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

   if 42 < y < 9.79999999999999967e51

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 72.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 9.79999999999999967e51 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\frac{\cos x}{\frac{1}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{if}\;y \leq 42:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (+
             0.16666666666666666
             (*
              y
              (*
               y
               (+
                0.008333333333333333
                (* (* y y) 0.0001984126984126984))))))))))
   (if (<= y 42.0) t_0 (if (<= y 9.8e+51) (/ (sinh y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	double tmp;
	if (y <= 42.0) {
		tmp = t_0;
	} else if (y <= 9.8e+51) {
		tmp = sinh(y) / y;
	} 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 = cos(x) * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    if (y <= 42.0d0) then
        tmp = t_0
    else if (y <= 9.8d+51) then
        tmp = sinh(y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	double tmp;
	if (y <= 42.0) {
		tmp = t_0;
	} else if (y <= 9.8e+51) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	tmp = 0
	if y <= 42.0:
		tmp = t_0
	elif y <= 9.8e+51:
		tmp = math.sinh(y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))
	tmp = 0.0
	if (y <= 42.0)
		tmp = t_0;
	elseif (y <= 9.8e+51)
		tmp = Float64(sinh(y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	tmp = 0.0;
	if (y <= 42.0)
		tmp = t_0;
	elseif (y <= 9.8e+51)
		tmp = sinh(y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 42.0], t$95$0, If[LessEqual[y, 9.8e+51], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 42 or 9.79999999999999967e51 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \cos x + {y}^{2} \cdot \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{120} \cdot \cos x\right)\right)} \]

   5. Simplified 93.7%

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

   if 42 < y < 9.79999999999999967e51

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 72.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 42.0)
   (*
    (cos x)
    (+
     1.0
     (* (* y y) (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))))
   (if (<= y 2e+75)
     (/ (sinh y) y)
     (* (cos x) (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	} else if (y <= 2e+75) {
		tmp = sinh(y) / y;
	} else {
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 42.0d0) then
        tmp = cos(x) * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0)))))
    else if (y <= 2d+75) then
        tmp = sinh(y) / y
    else
        tmp = cos(x) * (y * (y * (0.008333333333333333d0 * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = Math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	} else if (y <= 2e+75) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = Math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 42.0:
		tmp = math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))
	elif y <= 2e+75:
		tmp = math.sinh(y) / y
	else:
		tmp = math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 42.0)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333))))));
	elseif (y <= 2e+75)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(cos(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 42.0)
		tmp = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	elseif (y <= 2e+75)
		tmp = sinh(y) / y;
	else
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 42.0], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+75], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 42

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 90.6%

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

   7. Applied egg-rr 90.6%

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

   if 42 < y < 1.99999999999999985e75

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 68.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 1.99999999999999985e75 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \cos x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\cos x} \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.08:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.08)
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 3.9e+77)
     (* (/ (sinh y) y) (+ 1.0 (* x (* x -0.5))))
     (* (cos x) (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.08) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 3.9e+77) {
		tmp = (sinh(y) / y) * (1.0 + (x * (x * -0.5)));
	} else {
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.08d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 3.9d+77) then
        tmp = (sinh(y) / y) * (1.0d0 + (x * (x * (-0.5d0))))
    else
        tmp = cos(x) * (y * (y * (0.008333333333333333d0 * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.08) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 3.9e+77) {
		tmp = (Math.sinh(y) / y) * (1.0 + (x * (x * -0.5)));
	} else {
		tmp = Math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.08:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 3.9e+77:
		tmp = (math.sinh(y) / y) * (1.0 + (x * (x * -0.5)))
	else:
		tmp = math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.08)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 3.9e+77)
		tmp = Float64(Float64(sinh(y) / y) * Float64(1.0 + Float64(x * Float64(x * -0.5))));
	else
		tmp = Float64(cos(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.08)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 3.9e+77)
		tmp = (sinh(y) / y) * (1.0 + (x * (x * -0.5)));
	else
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.08], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+77], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0800000000000000017

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.5%

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

   5. Simplified 84.5%

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

   if 0.0800000000000000017 < y < 3.8999999999999998e77

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 71.0%

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

   5. Simplified 71.0%

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

   if 3.8999999999999998e77 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \cos x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\cos x} \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.08:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 42.0)
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 2e+75)
     (/ (sinh y) y)
     (* (cos x) (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 2e+75) {
		tmp = sinh(y) / y;
	} else {
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 42.0d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 2d+75) then
        tmp = sinh(y) / y
    else
        tmp = cos(x) * (y * (y * (0.008333333333333333d0 * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 2e+75) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = Math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 42.0:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 2e+75:
		tmp = math.sinh(y) / y
	else:
		tmp = math.cos(x) * (y * (y * (0.008333333333333333 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 42.0)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 2e+75)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(cos(x) * Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 42.0)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 2e+75)
		tmp = sinh(y) / y;
	else
		tmp = cos(x) * (y * (y * (0.008333333333333333 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 42.0], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+75], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 42

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.3%

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

   5. Simplified 84.3%

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

   if 42 < y < 1.99999999999999985e75

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 68.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 1.99999999999999985e75 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \cos x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\cos x} \]

      2. *-commutative N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \cos x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \cos x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\cos x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 42.0)
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 1.95e+152)
     (/ (sinh y) y)
     (* y (* y (* (cos x) 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.95e+152) {
		tmp = sinh(y) / y;
	} else {
		tmp = y * (y * (cos(x) * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 42.0d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 1.95d+152) then
        tmp = sinh(y) / y
    else
        tmp = y * (y * (cos(x) * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.95e+152) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = y * (y * (Math.cos(x) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 42.0:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 1.95e+152:
		tmp = math.sinh(y) / y
	else:
		tmp = y * (y * (math.cos(x) * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 42.0)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 1.95e+152)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(y * Float64(y * Float64(cos(x) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 42.0)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 1.95e+152)
		tmp = sinh(y) / y;
	else
		tmp = y * (y * (cos(x) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 42.0], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+152], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(y * N[(y * N[(N[Cos[x], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 42

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.3%

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

   5. Simplified 84.3%

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

   if 42 < y < 1.95000000000000006e152

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 75.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 1.95000000000000006e152 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. cos-lowering-cos.f64 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\cos x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00026:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\cos x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00026)
   (cos x)
   (if (<= y 1.95e+152)
     (/ (sinh y) y)
     (* y (* y (* (cos x) 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00026) {
		tmp = cos(x);
	} else if (y <= 1.95e+152) {
		tmp = sinh(y) / y;
	} else {
		tmp = y * (y * (cos(x) * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.00026d0) then
        tmp = cos(x)
    else if (y <= 1.95d+152) then
        tmp = sinh(y) / y
    else
        tmp = y * (y * (cos(x) * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00026) {
		tmp = Math.cos(x);
	} else if (y <= 1.95e+152) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = y * (y * (Math.cos(x) * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00026:
		tmp = math.cos(x)
	elif y <= 1.95e+152:
		tmp = math.sinh(y) / y
	else:
		tmp = y * (y * (math.cos(x) * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00026)
		tmp = cos(x);
	elseif (y <= 1.95e+152)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(y * Float64(y * Float64(cos(x) * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00026)
		tmp = cos(x);
	elseif (y <= 1.95e+152)
		tmp = sinh(y) / y;
	else
		tmp = y * (y * (cos(x) * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00026], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.95e+152], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(y * N[(y * N[(N[Cos[x], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.59999999999999977e-4

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 67.6%

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

   5. Simplified 67.6%

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

   if 2.59999999999999977e-4 < y < 1.95000000000000006e152

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 73.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   if 1.95000000000000006e152 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. cos-lowering-cos.f64 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00026:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\cos x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00065:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.00065) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00065) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.00065d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00065) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00065:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00065)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00065)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00065], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999997e-4

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 67.6%

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

   5. Simplified 67.6%

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

   if 6.4999999999999997e-4 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 77.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00065:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\\ t_1 := \left(y \cdot y\right) \cdot t\_0\\ \mathbf{if}\;y \leq 0.0019:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + t\_1 \cdot \left(y \cdot \left(t\_1 \cdot \left(y \cdot t\_0\right)\right)\right)}{1 + t\_1 \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))
        (t_1 (* (* y y) t_0)))
   (if (<= y 0.0019)
     (cos x)
     (if (<= y 7e+51)
       (/
        (+ 1.0 (* t_1 (* y (* t_1 (* y t_0)))))
        (+ 1.0 (* t_1 (+ (* y (* y 0.16666666666666666)) -1.0))))
       (if (<= y 6e+138)
         (*
          (* (* y y) (* y y))
          (+ 0.008333333333333333 (* (* x x) -0.004166666666666667)))
         (* y (* y (* 0.008333333333333333 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 0.0019) {
		tmp = cos(x);
	} else if (y <= 7e+51) {
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	} else if (y <= 6e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))
    t_1 = (y * y) * t_0
    if (y <= 0.0019d0) then
        tmp = cos(x)
    else if (y <= 7d+51) then
        tmp = (1.0d0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0d0 + (t_1 * ((y * (y * 0.16666666666666666d0)) + (-1.0d0))))
    else if (y <= 6d+138) then
        tmp = ((y * y) * (y * y)) * (0.008333333333333333d0 + ((x * x) * (-0.004166666666666667d0)))
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 0.0019) {
		tmp = Math.cos(x);
	} else if (y <= 7e+51) {
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	} else if (y <= 6e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333))
	t_1 = (y * y) * t_0
	tmp = 0
	if y <= 0.0019:
		tmp = math.cos(x)
	elif y <= 7e+51:
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)))
	elif y <= 6e+138:
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667))
	else:
		tmp = y * (y * (0.008333333333333333 * (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333)))
	t_1 = Float64(Float64(y * y) * t_0)
	tmp = 0.0
	if (y <= 0.0019)
		tmp = cos(x);
	elseif (y <= 7e+51)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(y * Float64(t_1 * Float64(y * t_0))))) / Float64(1.0 + Float64(t_1 * Float64(Float64(y * Float64(y * 0.16666666666666666)) + -1.0))));
	elseif (y <= 6e+138)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.004166666666666667)));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	t_1 = (y * y) * t_0;
	tmp = 0.0;
	if (y <= 0.0019)
		tmp = cos(x);
	elseif (y <= 7e+51)
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	elseif (y <= 6e+138)
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	else
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y, 0.0019], N[Cos[x], $MachinePrecision], If[LessEqual[y, 7e+51], N[(N[(1.0 + N[(t$95$1 * N[(y * N[(t$95$1 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+138], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 0.0019

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 67.6%

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

   5. Simplified 67.6%

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

   if 0.0019 < y < 7e51

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 15.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 10.7%

      \[\leadsto \color{blue}{1} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 + \color{blue}{\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} \]

      2. flip3-+ N/A

         \[\leadsto \frac{{1}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({1}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}^{3}\right), \color{blue}{\left(1 \cdot 1 + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right)\right)}\right) \]

   10. Applied egg-rr 31.5%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \frac{1}{6}\right), 1\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right), 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right), 1\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

       7. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

   13. Simplified 54.6%

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

   if 7e51 < y < 6.0000000000000002e138

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

       28. *-lowering-*.f64 60.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   11. Simplified 60.6%

       \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{240} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{240} \cdot {x}^{2}\right) \cdot {y}^{4} + \color{blue}{\frac{1}{120}} \cdot {y}^{4} \]

       2. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       5. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \color{blue}{\frac{-1}{240} \cdot {x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{240}\right)\right)\right) \]

       16. *-lowering-*.f64 70.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{240}\right)\right)\right) \]

   14. Simplified 70.6%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)} \]

   if 6.0000000000000002e138 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right) \]

       9. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right) \]

       10. cube-mult N/A

           \[\leadsto y \cdot \left({y}^{3} \cdot \frac{1}{120}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right) \]

       12. unpow3 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \]

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

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right) \]

       22. associate-*l* N/A

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

       23. *-commutative N/A

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

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

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

       25. *-commutative N/A

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

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right) \]

       28. *-lowering-*.f64 84.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right) \]

   11. Simplified 84.8%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0019:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)\right)}{1 + \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\\ t_1 := \left(y \cdot y\right) \cdot t\_0\\ \mathbf{if}\;y \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + t\_1 \cdot \left(y \cdot \left(t\_1 \cdot \left(y \cdot t\_0\right)\right)\right)}{1 + t\_1 \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))
        (t_1 (* (* y y) t_0)))
   (if (<= y 7e+51)
     (/
      (+ 1.0 (* t_1 (* y (* t_1 (* y t_0)))))
      (+ 1.0 (* t_1 (+ (* y (* y 0.16666666666666666)) -1.0))))
     (if (<= y 4e+138)
       (*
        (* (* y y) (* y y))
        (+ 0.008333333333333333 (* (* x x) -0.004166666666666667)))
       (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 7e+51) {
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	} else if (y <= 4e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))
    t_1 = (y * y) * t_0
    if (y <= 7d+51) then
        tmp = (1.0d0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0d0 + (t_1 * ((y * (y * 0.16666666666666666d0)) + (-1.0d0))))
    else if (y <= 4d+138) then
        tmp = ((y * y) * (y * y)) * (0.008333333333333333d0 + ((x * x) * (-0.004166666666666667d0)))
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 7e+51) {
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	} else if (y <= 4e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333))
	t_1 = (y * y) * t_0
	tmp = 0
	if y <= 7e+51:
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)))
	elif y <= 4e+138:
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667))
	else:
		tmp = y * (y * (0.008333333333333333 * (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333)))
	t_1 = Float64(Float64(y * y) * t_0)
	tmp = 0.0
	if (y <= 7e+51)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(y * Float64(t_1 * Float64(y * t_0))))) / Float64(1.0 + Float64(t_1 * Float64(Float64(y * Float64(y * 0.16666666666666666)) + -1.0))));
	elseif (y <= 4e+138)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.004166666666666667)));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	t_1 = (y * y) * t_0;
	tmp = 0.0;
	if (y <= 7e+51)
		tmp = (1.0 + (t_1 * (y * (t_1 * (y * t_0))))) / (1.0 + (t_1 * ((y * (y * 0.16666666666666666)) + -1.0)));
	elseif (y <= 4e+138)
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	else
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y, 7e+51], N[(N[(1.0 + N[(t$95$1 * N[(y * N[(t$95$1 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+138], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7e51

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 85.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 53.8%

      \[\leadsto \color{blue}{1} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 + \color{blue}{\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} \]

      2. flip3-+ N/A

         \[\leadsto \frac{{1}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left({1}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}^{3}\right), \color{blue}{\left(1 \cdot 1 + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right)\right)}\right) \]

   10. Applied egg-rr 40.1%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \frac{1}{6}\right), 1\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right), 1\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot y\right)\right), 1\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

       7. *-lowering-*.f64 43.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{120}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right), 1\right)\right)\right)\right) \]

   13. Simplified 43.6%

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

   if 7e51 < y < 4.0000000000000001e138

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 60.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

       28. *-lowering-*.f64 60.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   11. Simplified 60.6%

       \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{240} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{240} \cdot {x}^{2}\right) \cdot {y}^{4} + \color{blue}{\frac{1}{120}} \cdot {y}^{4} \]

       2. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       5. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \color{blue}{\frac{-1}{240} \cdot {x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{240}\right)\right)\right) \]

       16. *-lowering-*.f64 70.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{240}\right)\right)\right) \]

   14. Simplified 70.6%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)} \]

   if 4.0000000000000001e138 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right) \]

       9. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right) \]

       10. cube-mult N/A

           \[\leadsto y \cdot \left({y}^{3} \cdot \frac{1}{120}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right) \]

       12. unpow3 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \]

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

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right) \]

       22. associate-*l* N/A

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

       23. *-commutative N/A

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

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

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

       25. *-commutative N/A

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

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right) \]

       28. *-lowering-*.f64 84.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right) \]

   11. Simplified 84.8%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)\right)}{1 + \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\\ t_1 := \left(y \cdot y\right) \cdot t\_0\\ \mathbf{if}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{y \cdot \left(t\_1 \cdot \left(y \cdot t\_0\right)\right) + -1}{t\_1 + -1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))
        (t_1 (* (* y y) t_0)))
   (if (<= y 2e+75)
     (/ (+ (* y (* t_1 (* y t_0))) -1.0) (+ t_1 -1.0))
     (if (<= y 6e+138)
       (*
        (* (* y y) (* y y))
        (+ 0.008333333333333333 (* (* x x) -0.004166666666666667)))
       (* y (* y (* 0.008333333333333333 (* y y))))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 2e+75) {
		tmp = ((y * (t_1 * (y * t_0))) + -1.0) / (t_1 + -1.0);
	} else if (y <= 6e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))
    t_1 = (y * y) * t_0
    if (y <= 2d+75) then
        tmp = ((y * (t_1 * (y * t_0))) + (-1.0d0)) / (t_1 + (-1.0d0))
    else if (y <= 6d+138) then
        tmp = ((y * y) * (y * y)) * (0.008333333333333333d0 + ((x * x) * (-0.004166666666666667d0)))
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	double t_1 = (y * y) * t_0;
	double tmp;
	if (y <= 2e+75) {
		tmp = ((y * (t_1 * (y * t_0))) + -1.0) / (t_1 + -1.0);
	} else if (y <= 6e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333))
	t_1 = (y * y) * t_0
	tmp = 0
	if y <= 2e+75:
		tmp = ((y * (t_1 * (y * t_0))) + -1.0) / (t_1 + -1.0)
	elif y <= 6e+138:
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667))
	else:
		tmp = y * (y * (0.008333333333333333 * (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333)))
	t_1 = Float64(Float64(y * y) * t_0)
	tmp = 0.0
	if (y <= 2e+75)
		tmp = Float64(Float64(Float64(y * Float64(t_1 * Float64(y * t_0))) + -1.0) / Float64(t_1 + -1.0));
	elseif (y <= 6e+138)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.004166666666666667)));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + (y * (y * 0.008333333333333333));
	t_1 = (y * y) * t_0;
	tmp = 0.0;
	if (y <= 2e+75)
		tmp = ((y * (t_1 * (y * t_0))) + -1.0) / (t_1 + -1.0);
	elseif (y <= 6e+138)
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	else
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y, 2e+75], N[(N[(N[(y * N[(t$95$1 * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+138], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999985e75

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 84.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 52.6%

      \[\leadsto \color{blue}{1} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 + \color{blue}{\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right) + \color{blue}{1} \]

      3. flip-+ N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot 1}{\color{blue}{\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right) - 1}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) - 1 \cdot 1\right), \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right) - 1\right)}\right) \]

   10. Applied egg-rr 41.2%

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

   if 1.99999999999999985e75 < y < 6.0000000000000002e138

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

       28. *-lowering-*.f64 73.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   11. Simplified 73.3%

       \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{240} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{240} \cdot {x}^{2}\right) \cdot {y}^{4} + \color{blue}{\frac{1}{120}} \cdot {y}^{4} \]

       2. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       5. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \color{blue}{\frac{-1}{240} \cdot {x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{240}\right)\right)\right) \]

       16. *-lowering-*.f64 80.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{240}\right)\right)\right) \]

   14. Simplified 80.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)} \]

   if 6.0000000000000002e138 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right) \]

       9. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right) \]

       10. cube-mult N/A

           \[\leadsto y \cdot \left({y}^{3} \cdot \frac{1}{120}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right) \]

       12. unpow3 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \]

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

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right) \]

       22. associate-*l* N/A

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

       23. *-commutative N/A

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

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

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

       25. *-commutative N/A

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

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right) \]

       28. *-lowering-*.f64 84.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right) \]

   11. Simplified 84.8%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{y \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right) \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right) + -1}{\left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right) + -1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+103}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.3e+103)
   (+
    1.0
    (*
     (* y y)
     (+
      0.16666666666666666
      (* (* y y) (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))
   (if (<= x 3.3e+183)
     (*
      (* (* y y) (* y y))
      (+ 0.008333333333333333 (* (* x x) -0.004166666666666667)))
     (+ 1.0 (* x (* x (+ -0.5 (* x (* x 0.041666666666666664)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.3e+103) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))));
	} else if (x <= 3.3e+183) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (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 <= 5.3d+103) then
        tmp = 1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))
    else if (x <= 3.3d+183) then
        tmp = ((y * y) * (y * y)) * (0.008333333333333333d0 + ((x * x) * (-0.004166666666666667d0)))
    else
        tmp = 1.0d0 + (x * (x * ((-0.5d0) + (x * (x * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.3e+103) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))));
	} else if (x <= 3.3e+183) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.3e+103:
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))
	elif x <= 3.3e+183:
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667))
	else:
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.3e+103)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984)))))));
	elseif (x <= 3.3e+183)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.004166666666666667)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.3e+103)
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))));
	elseif (x <= 3.3e+183)
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	else
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.3e+103], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+183], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.29999999999999969e103

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \sinh y}{\color{blue}{y}} \]

      2. div-inv N/A

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

      3. sinh-def N/A

         \[\leadsto \left(\cos x \cdot \frac{e^{y} - e^{\mathsf{neg}\left(y\right)}}{2}\right) \cdot \frac{1}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)}{2} \cdot \frac{\color{blue}{1}}{y} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}{\color{blue}{2}} \]

      6. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\left(\cos x \cdot \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right)\right) \cdot \frac{1}{y}}}} \]

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

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

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

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

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

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

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

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

      15. div-inv N/A

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

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

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

      17. sinh-undef N/A

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

      18. associate-/r* N/A

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{\cos x}{y \cdot \frac{0.5}{\sinh y}}}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

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

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. sinh-def N/A

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

      14. clear-num N/A

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

      15. frac-times N/A

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

      16. clear-num N/A

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

      17. sinh-def N/A

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\cos x}{\frac{y}{\sinh y}}}}} \]

   7. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 91.5%

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

   9. Simplified 91.5%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   11. Step-by-step derivation

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

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

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

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

       3. unpow2 N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

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

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

       14. *-lowering-*.f64 64.9%

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

   12. Simplified 64.9%

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

   if 5.29999999999999969e103 < x < 3.3000000000000001e183

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 29.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

       28. *-lowering-*.f64 28.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   11. Simplified 28.8%

       \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{240} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{240} \cdot {x}^{2}\right) \cdot {y}^{4} + \color{blue}{\frac{1}{120}} \cdot {y}^{4} \]

       2. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       5. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \color{blue}{\frac{-1}{240} \cdot {x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{240}\right)\right)\right) \]

       16. *-lowering-*.f64 29.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{240}\right)\right)\right) \]

   14. Simplified 29.7%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)} \]

   if 3.3000000000000001e183 < x

   1. Initial program 99.9%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 79.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 30.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 30.1%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

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

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

       14. *-lowering-*.f64 30.1%

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

   11. Simplified 30.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.008333333333333333 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + t\_0\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.008333333333333333 (* y y))))
   (if (<= y 0.25)
     (+ 1.0 (* y (* y (+ 0.16666666666666666 t_0))))
     (if (<= y 2.4e+138)
       (*
        (* (* y y) (* y y))
        (+ 0.008333333333333333 (* (* x x) -0.004166666666666667)))
       (* y (* y t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.008333333333333333 * (y * y);
	double tmp;
	if (y <= 0.25) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + t_0)));
	} else if (y <= 2.4e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * 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 = 0.008333333333333333d0 * (y * y)
    if (y <= 0.25d0) then
        tmp = 1.0d0 + (y * (y * (0.16666666666666666d0 + t_0)))
    else if (y <= 2.4d+138) then
        tmp = ((y * y) * (y * y)) * (0.008333333333333333d0 + ((x * x) * (-0.004166666666666667d0)))
    else
        tmp = y * (y * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.008333333333333333 * (y * y);
	double tmp;
	if (y <= 0.25) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + t_0)));
	} else if (y <= 2.4e+138) {
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	} else {
		tmp = y * (y * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.008333333333333333 * (y * y)
	tmp = 0
	if y <= 0.25:
		tmp = 1.0 + (y * (y * (0.16666666666666666 + t_0)))
	elif y <= 2.4e+138:
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667))
	else:
		tmp = y * (y * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.008333333333333333 * Float64(y * y))
	tmp = 0.0
	if (y <= 0.25)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + t_0))));
	elseif (y <= 2.4e+138)
		tmp = Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.004166666666666667)));
	else
		tmp = Float64(y * Float64(y * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.008333333333333333 * (y * y);
	tmp = 0.0;
	if (y <= 0.25)
		tmp = 1.0 + (y * (y * (0.16666666666666666 + t_0)));
	elseif (y <= 2.4e+138)
		tmp = ((y * y) * (y * y)) * (0.008333333333333333 + ((x * x) * -0.004166666666666667));
	else
		tmp = y * (y * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.25], N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+138], N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.004166666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.25

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 57.0%

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

   8. Simplified 57.0%

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

   if 0.25 < y < 2.4000000000000001e138

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 51.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 48.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 48.0%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       10. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

       28. *-lowering-*.f64 48.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   11. Simplified 48.0%

       \[\leadsto \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{240} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{240} \cdot {x}^{2}\right) \cdot {y}^{4} + \color{blue}{\frac{1}{120}} \cdot {y}^{4} \]

       2. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{120}\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       5. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240} \cdot {x}^{2}} + \frac{1}{120}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{-1}{240}} \cdot {x}^{2} + \frac{1}{120}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{-1}{240} \cdot \color{blue}{{x}^{2}} + \frac{1}{120}\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \color{blue}{\frac{-1}{240} \cdot {x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{240} \cdot {x}^{2}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{240}}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{240}\right)\right)\right) \]

       16. *-lowering-*.f64 51.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{240}\right)\right)\right) \]

   14. Simplified 51.5%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)} \]

   if 2.4000000000000001e138 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 84.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right) \]

       9. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right) \]

       10. cube-mult N/A

           \[\leadsto y \cdot \left({y}^{3} \cdot \frac{1}{120}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right) \]

       12. unpow3 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \]

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

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

       16. associate-*r* N/A

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

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right) \]

       22. associate-*l* N/A

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

       23. *-commutative N/A

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

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

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

       25. *-commutative N/A

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

       26. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right) \]

       28. *-lowering-*.f64 84.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right) \]

   11. Simplified 84.8%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.004166666666666667\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+167}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.8e+167)
   (+ 1.0 (* y (* y (+ 0.16666666666666666 (* 0.008333333333333333 (* y y))))))
   (if (<= x 3.2e+184)
     (+ 1.0 (* -0.5 (* x x)))
     (+ 1.0 (* x (* x (+ -0.5 (* x (* x 0.041666666666666664)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+167) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (0.008333333333333333 * (y * y)))));
	} else if (x <= 3.2e+184) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (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 <= 4.8d+167) then
        tmp = 1.0d0 + (y * (y * (0.16666666666666666d0 + (0.008333333333333333d0 * (y * y)))))
    else if (x <= 3.2d+184) then
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    else
        tmp = 1.0d0 + (x * (x * ((-0.5d0) + (x * (x * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+167) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (0.008333333333333333 * (y * y)))));
	} else if (x <= 3.2e+184) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.8e+167:
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (0.008333333333333333 * (y * y)))))
	elif x <= 3.2e+184:
		tmp = 1.0 + (-0.5 * (x * x))
	else:
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.8e+167)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(0.008333333333333333 * Float64(y * y))))));
	elseif (x <= 3.2e+184)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.8e+167)
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (0.008333333333333333 * (y * y)))));
	elseif (x <= 3.2e+184)
		tmp = 1.0 + (-0.5 * (x * x));
	else
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.8e+167], N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+184], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.79999999999999998e167

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 61.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 61.9%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   if 4.79999999999999998e167 < x < 3.19999999999999983e184

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 35.4%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 35.4%

      \[\leadsto \color{blue}{\cos x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{2}\right)\right) \]

      5. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]

   if 3.19999999999999983e184 < x

   1. Initial program 99.9%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 79.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 30.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 30.1%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 30.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 30.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+167}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+167}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.8e+167)
   (+ 1.0 (* y (* y (* 0.008333333333333333 (* y y)))))
   (if (<= x 3.2e+184)
     (+ 1.0 (* -0.5 (* x x)))
     (+ 1.0 (* x (* x (+ -0.5 (* x (* x 0.041666666666666664)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+167) {
		tmp = 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
	} else if (x <= 3.2e+184) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (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 <= 4.8d+167) then
        tmp = 1.0d0 + (y * (y * (0.008333333333333333d0 * (y * y))))
    else if (x <= 3.2d+184) then
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    else
        tmp = 1.0d0 + (x * (x * ((-0.5d0) + (x * (x * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.8e+167) {
		tmp = 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
	} else if (x <= 3.2e+184) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.8e+167:
		tmp = 1.0 + (y * (y * (0.008333333333333333 * (y * y))))
	elif x <= 3.2e+184:
		tmp = 1.0 + (-0.5 * (x * x))
	else:
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.8e+167)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))));
	elseif (x <= 3.2e+184)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.8e+167)
		tmp = 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
	elseif (x <= 3.2e+184)
		tmp = 1.0 + (-0.5 * (x * x));
	else
		tmp = 1.0 + (x * (x * (-0.5 + (x * (x * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.8e+167], N[(1.0 + N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+184], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.79999999999999998e167

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 87.1%

      \[\leadsto \cos x \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       13. unpow3 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

       15. associate-*r* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

       17. associate-*r* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       19. unpow3 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       21. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

       23. associate-*l* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

       26. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

       27. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

   11. Simplified 61.5%

       \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   if 4.79999999999999998e167 < x < 3.19999999999999983e184

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 35.4%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 35.4%

      \[\leadsto \color{blue}{\cos x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{2}\right)\right) \]

      5. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]

   if 3.19999999999999983e184 < x

   1. Initial program 99.9%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 79.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 30.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 30.1%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{1}{24}\right)}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 30.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 30.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+167}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+184}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(-0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+71}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.25)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (if (<= y 1.85e+71)
     (+ 1.0 (* -0.5 (* x x)))
     (* y (* y (* 0.008333333333333333 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.25) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else if (y <= 1.85e+71) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.25d0) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else if (y <= 1.85d+71) then
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.25) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else if (y <= 1.85e+71) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.25:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	elif y <= 1.85e+71:
		tmp = 1.0 + (-0.5 * (x * x))
	else:
		tmp = y * (y * (0.008333333333333333 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.25)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	elseif (y <= 1.85e+71)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.25)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	elseif (y <= 1.85e+71)
		tmp = 1.0 + (-0.5 * (x * x));
	else
		tmp = y * (y * (0.008333333333333333 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.25], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+71], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.25

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 84.5%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 53.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 53.4%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 0.25 < y < 1.85e71

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 4.7%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 4.7%

      \[\leadsto \color{blue}{\cos x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{2}\right)\right) \]

      5. *-lowering-*.f64 27.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 27.1%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]

   if 1.85e71 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 79.8%

      \[\leadsto \color{blue}{1} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right) \]

       9. unpow2 N/A

          \[\leadsto y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right) \]

       10. cube-mult N/A

           \[\leadsto y \cdot \left({y}^{3} \cdot \frac{1}{120}\right) \]

       11. *-commutative N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right) \]

       12. unpow3 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right) \]

       16. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right) \]

       18. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       20. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right) \]

       22. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       24. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       26. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right) \]

       27. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right) \]

       28. *-lowering-*.f64 79.8%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right) \]

   11. Simplified 79.8%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+71}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+114}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 0.25) t_0 (if (<= y 1.75e+114) (+ 1.0 (* -0.5 (* x x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.25) {
		tmp = t_0;
	} else if (y <= 1.75e+114) {
		tmp = 1.0 + (-0.5 * (x * x));
	} 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 + (0.16666666666666666d0 * (y * y))
    if (y <= 0.25d0) then
        tmp = t_0
    else if (y <= 1.75d+114) then
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 0.25) {
		tmp = t_0;
	} else if (y <= 1.75e+114) {
		tmp = 1.0 + (-0.5 * (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 0.25:
		tmp = t_0
	elif y <= 1.75e+114:
		tmp = 1.0 + (-0.5 * (x * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 0.25)
		tmp = t_0;
	elseif (y <= 1.75e+114)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 0.25)
		tmp = t_0;
	elseif (y <= 1.75e+114)
		tmp = 1.0 + (-0.5 * (x * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.25], t$95$0, If[LessEqual[y, 1.75e+114], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.25 or 1.75e114 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 55.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 0.25 < y < 1.75e114

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 4.0%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 4.0%

      \[\leadsto \color{blue}{\cos x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{2}\right)\right) \]

      5. *-lowering-*.f64 23.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{2}\right)\right) \]

   8. Simplified 23.6%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.25:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+114}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* y (* y (* 0.008333333333333333 (* y y))))))

C

double code(double x, double y) {
	return 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (y * (y * (0.008333333333333333d0 * (y * y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
}

Python

def code(x, y):
	return 1.0 + (y * (y * (0.008333333333333333 * (y * y))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (y * (y * (0.008333333333333333 * (y * y))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(y * N[(y * N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

   5. associate-*r* N/A

      \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

   6. distribute-rgt-out N/A

      \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

   7. +-commutative N/A

      \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

   8. associate-*l* N/A

      \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

   10. distribute-lft-in N/A

       \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

5. Simplified 87.0%

   \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right)\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

   2. pow-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 86.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

8. Simplified 86.3%

   \[\leadsto \cos x \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)}\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{120} \cdot {y}^{4}} \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

    3. pow-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

    11. cube-mult N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left({y}^{3} \cdot \frac{1}{120}\right)\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

    13. unpow3 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

    15. associate-*r* N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

    17. associate-*r* N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

    18. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

    19. unpow3 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

    20. *-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{3} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right) \]

    21. cube-mult N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120}\right)\right)\right) \]

    22. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot {y}^{2}\right) \cdot \frac{1}{120}\right)\right)\right) \]

    23. associate-*l* N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{120}\right)}\right)\right)\right) \]

    24. *-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

    25. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

    26. *-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

    27. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

11. Simplified 57.7%

    \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

12. Final simplification 57.7%

    \[\leadsto 1 + y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot y\right)\right)\right) \]
13. Add Preprocessing

Alternative 21: 37.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 42.0) 1.0 (* y (* y 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = 1.0;
	} else {
		tmp = y * (y * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 42.0d0) then
        tmp = 1.0d0
    else
        tmp = y * (y * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 42.0) {
		tmp = 1.0;
	} else {
		tmp = y * (y * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 42.0:
		tmp = 1.0
	else:
		tmp = y * (y * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 42.0)
		tmp = 1.0;
	else
		tmp = Float64(y * Float64(y * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 42.0)
		tmp = 1.0;
	else
		tmp = y * (y * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 42.0], 1.0, N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 42

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 67.3%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\cos x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 39.1%

      \[\leadsto \color{blue}{1} \]

   if 42 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 45.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 45.5%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 38.4%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\frac{1}{6}} \]

       2. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \frac{1}{6} \]

       3. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{y}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       7. *-lowering-*.f64 38.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right) \]

   11. Simplified 38.4%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 46.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.16666666666666666d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

   2. *-lft-identity N/A

      \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

   3. distribute-rgt-in N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 74.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

5. Simplified 74.6%

   \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   4. *-lowering-*.f64 49.4%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

8. Simplified 49.4%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
9. Add Preprocessing

Alternative 23: 28.3% 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 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 51.3%

      \[\leadsto \mathsf{cos.f64}\left(x\right) \]

5. Simplified 51.3%

   \[\leadsto \color{blue}{\cos x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 29.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))