Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 15.1s

Alternatives: 25

Speedup: 1.6×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* N/A

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

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

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

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

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

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

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

   5. sinh-lowering-sinh.f64 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (sin x)
          (/
           (*
            y
            (+
             1.0
             (*
              (* y y)
              (+
               0.16666666666666666
               (*
                y
                (*
                 y
                 (+
                  0.008333333333333333
                  (* (* y y) 0.0001984126984126984))))))))
           x))))
   (if (<= y 10.2)
     t_0
     (if (<= y 6.5e+27)
       (/ (* (sinh y) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	double tmp;
	if (y <= 10.2) {
		tmp = t_0;
	} else if (y <= 6.5e+27) {
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / 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 = sin(x) * ((y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) / x)
    if (y <= 10.2d0) then
        tmp = t_0
    else if (y <= 6.5d+27) then
        tmp = (sinh(y) * (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	double tmp;
	if (y <= 10.2) {
		tmp = t_0;
	} else if (y <= 6.5e+27) {
		tmp = (Math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x)
	tmp = 0
	if y <= 10.2:
		tmp = t_0
	elif y <= 6.5e+27:
		tmp = (math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) / x))
	tmp = 0.0
	if (y <= 10.2)
		tmp = t_0;
	elseif (y <= 6.5e+27)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * ((y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / x);
	tmp = 0.0;
	if (y <= 10.2)
		tmp = t_0;
	elseif (y <= 6.5e+27)
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[(y * 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] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 10.2], t$95$0, If[LessEqual[y, 6.5e+27], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 10.199999999999999 or 6.5000000000000005e27 < y

   1. Initial program 88.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 97.2%

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

   7. Simplified 97.2%

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

   if 10.199999999999999 < y < 6.5000000000000005e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\sin x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\\ \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(\frac{1}{x} + \frac{y}{\frac{x}{y}} \cdot t\_0\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot t\_0\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))
   (if (<= y 10.2)
     (* (sin x) (* y (+ (/ 1.0 x) (* (/ y (/ x y)) t_0))))
     (if (<= y 1.15e+62)
       (/ (* (sinh y) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))) x)
       (/ (* y (* (sin x) (+ 1.0 (* y (* y t_0))))) x)))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333);
	double tmp;
	if (y <= 10.2) {
		tmp = sin(x) * (y * ((1.0 / x) + ((y / (x / y)) * t_0)));
	} else if (y <= 1.15e+62) {
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = (y * (sin(x) * (1.0 + (y * (y * t_0))))) / x;
	}
	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)
    if (y <= 10.2d0) then
        tmp = sin(x) * (y * ((1.0d0 / x) + ((y / (x / y)) * t_0)))
    else if (y <= 1.15d+62) then
        tmp = (sinh(y) * (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / x
    else
        tmp = (y * (sin(x) * (1.0d0 + (y * (y * t_0))))) / x
    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 tmp;
	if (y <= 10.2) {
		tmp = Math.sin(x) * (y * ((1.0 / x) + ((y / (x / y)) * t_0)));
	} else if (y <= 1.15e+62) {
		tmp = (Math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = (y * (Math.sin(x) * (1.0 + (y * (y * t_0))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333)
	tmp = 0
	if y <= 10.2:
		tmp = math.sin(x) * (y * ((1.0 / x) + ((y / (x / y)) * t_0)))
	elif y <= 1.15e+62:
		tmp = (math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x
	else:
		tmp = (y * (math.sin(x) * (1.0 + (y * (y * t_0))))) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))
	tmp = 0.0
	if (y <= 10.2)
		tmp = Float64(sin(x) * Float64(y * Float64(Float64(1.0 / x) + Float64(Float64(y / Float64(x / y)) * t_0))));
	elseif (y <= 1.15e+62)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * Float64(1.0 + Float64(y * Float64(y * t_0))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333);
	tmp = 0.0;
	if (y <= 10.2)
		tmp = sin(x) * (y * ((1.0 / x) + ((y / (x / y)) * t_0)));
	elseif (y <= 1.15e+62)
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	else
		tmp = (y * (sin(x) * (1.0 + (y * (y * t_0))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 10.2], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(N[(1.0 / x), $MachinePrecision] + N[(N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 10.199999999999999

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

      14. +-commutative N/A

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

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

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

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

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

      17. unpow2 N/A

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

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

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

   7. Simplified 92.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 92.9%

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

   9. Applied egg-rr 92.9%

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

   if 10.199999999999999 < y < 1.14999999999999992e62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

   if 1.14999999999999992e62 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(\frac{1}{x} + \frac{y}{\frac{x}{y}} \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10.2)
   (* (/ y (/ x (sin x))) (+ 1.0 (* y (* y 0.16666666666666666))))
   (if (<= y 1.25e+62)
     (/ (* (sinh y) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))) x)
     (/
      (*
       y
       (*
        (sin x)
        (+
         1.0
         (*
          y
          (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10.2) {
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 1.25e+62) {
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 10.2d0) then
        tmp = (y / (x / sin(x))) * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    else if (y <= 1.25d+62) then
        tmp = (sinh(y) * (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / x
    else
        tmp = (y * (sin(x) * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 10.2) {
		tmp = (y / (x / Math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 1.25e+62) {
		tmp = (Math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = (y * (Math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 10.2:
		tmp = (y / (x / math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)))
	elif y <= 1.25e+62:
		tmp = (math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x
	else:
		tmp = (y * (math.sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10.2)
		tmp = Float64(Float64(y / Float64(x / sin(x))) * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	elseif (y <= 1.25e+62)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / x);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 10.2)
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	elseif (y <= 1.25e+62)
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	else
		tmp = (y * (sin(x) * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10.2], N[(N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+62], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 10.199999999999999

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 86.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 86.7%

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

   9. Applied egg-rr 86.7%

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

   if 10.199999999999999 < y < 1.25000000000000007e62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

   if 1.25000000000000007e62 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8000:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8000.0)
   (* (/ y (/ x (sin x))) (+ 1.0 (* y (* y 0.16666666666666666))))
   (if (<= y 4e+49)
     (sinh y)
     (if (<= y 1.25e+103)
       (*
        (*
         y
         (+
          1.0
          (*
           (* y y)
           (+
            0.16666666666666666
            (*
             y
             (*
              y
              (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
        (+ 1.0 (* (* x x) -0.16666666666666666)))
       (/ (* (* y (* (sin x) y)) (* y 0.16666666666666666)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8000.0) {
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 4e+49) {
		tmp = sinh(y);
	} else if (y <= 1.25e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8000.0d0) then
        tmp = (y / (x / sin(x))) * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    else if (y <= 4d+49) then
        tmp = sinh(y)
    else if (y <= 1.25d+103) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8000.0) {
		tmp = (y / (x / Math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 4e+49) {
		tmp = Math.sinh(y);
	} else if (y <= 1.25e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (Math.sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8000.0:
		tmp = (y / (x / math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)))
	elif y <= 4e+49:
		tmp = math.sinh(y)
	elif y <= 1.25e+103:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = ((y * (math.sin(x) * y)) * (y * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8000.0)
		tmp = Float64(Float64(y / Float64(x / sin(x))) * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	elseif (y <= 4e+49)
		tmp = sinh(y);
	elseif (y <= 1.25e+103)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(Float64(y * Float64(sin(x) * y)) * Float64(y * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8000.0)
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	elseif (y <= 4e+49)
		tmp = sinh(y);
	elseif (y <= 1.25e+103)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8000.0], N[(N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+49], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 1.25e+103], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8e3

   1. Initial program 84.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 86.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 86.3%

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

   9. Applied egg-rr 86.3%

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

   if 8e3 < y < 3.99999999999999979e49

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

   7. Simplified 58.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 58.3%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\sinh y} \]

   if 3.99999999999999979e49 < y < 1.25e103

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

   10. Simplified 72.7%

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

   if 1.25e103 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

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

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

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

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

      19. unpow2 N/A

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

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

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

      21. *-lowering-*.f64 83.9%

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

   10. Simplified 83.9%

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(\left(\sin x \cdot y\right) \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8000:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8000:\\ \;\;\;\;y \cdot \left(\left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8000.0)
   (* y (* (+ 1.0 (* (* y y) 0.16666666666666666)) (/ (sin x) x)))
   (if (<= y 1.5e+49)
     (sinh y)
     (if (<= y 1.5e+103)
       (*
        (*
         y
         (+
          1.0
          (*
           (* y y)
           (+
            0.16666666666666666
            (*
             y
             (*
              y
              (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
        (+ 1.0 (* (* x x) -0.16666666666666666)))
       (/ (* (* y (* (sin x) y)) (* y 0.16666666666666666)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8000.0) {
		tmp = y * ((1.0 + ((y * y) * 0.16666666666666666)) * (sin(x) / x));
	} else if (y <= 1.5e+49) {
		tmp = sinh(y);
	} else if (y <= 1.5e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8000.0d0) then
        tmp = y * ((1.0d0 + ((y * y) * 0.16666666666666666d0)) * (sin(x) / x))
    else if (y <= 1.5d+49) then
        tmp = sinh(y)
    else if (y <= 1.5d+103) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8000.0) {
		tmp = y * ((1.0 + ((y * y) * 0.16666666666666666)) * (Math.sin(x) / x));
	} else if (y <= 1.5e+49) {
		tmp = Math.sinh(y);
	} else if (y <= 1.5e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (Math.sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8000.0:
		tmp = y * ((1.0 + ((y * y) * 0.16666666666666666)) * (math.sin(x) / x))
	elif y <= 1.5e+49:
		tmp = math.sinh(y)
	elif y <= 1.5e+103:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = ((y * (math.sin(x) * y)) * (y * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8000.0)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)) * Float64(sin(x) / x)));
	elseif (y <= 1.5e+49)
		tmp = sinh(y);
	elseif (y <= 1.5e+103)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(Float64(y * Float64(sin(x) * y)) * Float64(y * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8000.0)
		tmp = y * ((1.0 + ((y * y) * 0.16666666666666666)) * (sin(x) / x));
	elseif (y <= 1.5e+49)
		tmp = sinh(y);
	elseif (y <= 1.5e+103)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8000.0], N[(y * N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+49], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 1.5e+103], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8e3

   1. Initial program 84.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 86.3%

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

   if 8e3 < y < 1.5000000000000001e49

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

   7. Simplified 58.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 58.3%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{\sinh y} \]

   if 1.5000000000000001e49 < y < 1.5e103

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

   10. Simplified 72.7%

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

   if 1.5e103 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

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

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

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

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

      19. unpow2 N/A

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

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

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

      21. *-lowering-*.f64 83.9%

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

   10. Simplified 83.9%

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(\left(\sin x \cdot y\right) \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8000:\\ \;\;\;\;y \cdot \left(\left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \frac{\sin x}{x}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (* y (/ (sin x) x))
   (if (<= y 1e+49)
     (sinh y)
     (if (<= y 1.1e+103)
       (*
        (*
         y
         (+
          1.0
          (*
           (* y y)
           (+
            0.16666666666666666
            (*
             y
             (*
              y
              (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
        (+ 1.0 (* (* x x) -0.16666666666666666)))
       (/ (* (* y (* (sin x) y)) (* y 0.16666666666666666)) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (sin(x) / x);
	} else if (y <= 1e+49) {
		tmp = sinh(y);
	} else if (y <= 1.1e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0065d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 1d+49) then
        tmp = sinh(y)
    else if (y <= 1.1d+103) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 1e+49) {
		tmp = Math.sinh(y);
	} else if (y <= 1.1e+103) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = ((y * (Math.sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = y * (math.sin(x) / x)
	elif y <= 1e+49:
		tmp = math.sinh(y)
	elif y <= 1.1e+103:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = ((y * (math.sin(x) * y)) * (y * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 1e+49)
		tmp = sinh(y);
	elseif (y <= 1.1e+103)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(Float64(y * Float64(sin(x) * y)) * Float64(y * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0065)
		tmp = y * (sin(x) / x);
	elseif (y <= 1e+49)
		tmp = sinh(y);
	elseif (y <= 1.1e+103)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+49], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 1.1e+103], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 0.0064999999999999997

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 86.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. sin-lowering-sin.f64 67.9%

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

   10. Simplified 67.9%

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

   if 0.0064999999999999997 < y < 9.99999999999999946e48

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

   7. Simplified 50.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 50.2%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\sinh y} \]

   if 9.99999999999999946e48 < y < 1.09999999999999996e103

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

   10. Simplified 72.7%

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

   if 1.09999999999999996e103 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 83.9%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-/l* N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

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

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

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

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

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

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

      19. unpow2 N/A

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

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

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

      21. *-lowering-*.f64 83.9%

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

   10. Simplified 83.9%

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(\left(\sin x \cdot y\right) \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+134}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (* y (/ (sin x) x))
   (if (<= y 2.5e+49)
     (sinh y)
     (if (<= y 2.5e+134)
       (*
        (*
         y
         (+
          1.0
          (*
           (* y y)
           (+
            0.16666666666666666
            (*
             y
             (*
              y
              (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
        (+ 1.0 (* (* x x) -0.16666666666666666)))
       (* (* (sin x) (/ (* y y) x)) (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (sin(x) / x);
	} else if (y <= 2.5e+49) {
		tmp = sinh(y);
	} else if (y <= 2.5e+134) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (sin(x) * ((y * y) / x)) * (y * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0065d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 2.5d+49) then
        tmp = sinh(y)
    else if (y <= 2.5d+134) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = (sin(x) * ((y * y) / x)) * (y * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 2.5e+49) {
		tmp = Math.sinh(y);
	} else if (y <= 2.5e+134) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (Math.sin(x) * ((y * y) / x)) * (y * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = y * (math.sin(x) / x)
	elif y <= 2.5e+49:
		tmp = math.sinh(y)
	elif y <= 2.5e+134:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = (math.sin(x) * ((y * y) / x)) * (y * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 2.5e+49)
		tmp = sinh(y);
	elseif (y <= 2.5e+134)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(sin(x) * Float64(Float64(y * y) / x)) * Float64(y * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0065)
		tmp = y * (sin(x) / x);
	elseif (y <= 2.5e+49)
		tmp = sinh(y);
	elseif (y <= 2.5e+134)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = (sin(x) * ((y * y) / x)) * (y * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+49], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 2.5e+134], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 0.0064999999999999997

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 86.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. sin-lowering-sin.f64 67.9%

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

   10. Simplified 67.9%

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

   if 0.0064999999999999997 < y < 2.5000000000000002e49

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

   7. Simplified 50.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 50.2%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\sinh y} \]

   if 2.5000000000000002e49 < y < 2.4999999999999999e134

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 73.7%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]

   if 2.4999999999999999e134 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 94.4%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+134}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10.2)
   (* (/ y (/ x (sin x))) (+ 1.0 (* y (* y 0.16666666666666666))))
   (if (<= y 1.05e+103)
     (/ (* (sinh y) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))) x)
     (/ (* (* y (* (sin x) y)) (* y 0.16666666666666666)) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10.2) {
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 1.05e+103) {
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 10.2d0) then
        tmp = (y / (x / sin(x))) * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    else if (y <= 1.05d+103) then
        tmp = (sinh(y) * (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / x
    else
        tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 10.2) {
		tmp = (y / (x / Math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	} else if (y <= 1.05e+103) {
		tmp = (Math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = ((y * (Math.sin(x) * y)) * (y * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 10.2:
		tmp = (y / (x / math.sin(x))) * (1.0 + (y * (y * 0.16666666666666666)))
	elif y <= 1.05e+103:
		tmp = (math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x
	else:
		tmp = ((y * (math.sin(x) * y)) * (y * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10.2)
		tmp = Float64(Float64(y / Float64(x / sin(x))) * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	elseif (y <= 1.05e+103)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / x);
	else
		tmp = Float64(Float64(Float64(y * Float64(sin(x) * y)) * Float64(y * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 10.2)
		tmp = (y / (x / sin(x))) * (1.0 + (y * (y * 0.16666666666666666)));
	elseif (y <= 1.05e+103)
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	else
		tmp = ((y * (sin(x) * y)) * (y * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10.2], N[(N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+103], N[(N[(N[Sinh[y], $MachinePrecision] * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 10.199999999999999

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 86.7%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \left(\frac{\sin x}{x} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(y \cdot \frac{\sin x}{x}\right) \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{\sin x}{x}\right), \color{blue}{\left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{x}{\sin x}}\right), \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{x}{\sin x}}\right), \left(\color{blue}{1} + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x}{\sin x}\right)\right), \left(\color{blue}{1} + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \sin x\right)\right), \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \left(1 + \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot \left(y \cdot y\right)\right)}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{y}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 86.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   9. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   if 10.199999999999999 < y < 1.0500000000000001e103

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{sinh.f64}\left(y\right)\right), x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

      6. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 75.0%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \sinh y}{x} \]

   if 1.0500000000000001e103 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 83.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 83.9%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\sin x \cdot \left(y \cdot y\right)}{x} \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right) \]

       2. associate-*l/ N/A

          \[\leadsto \frac{\left(\sin x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{6} \cdot y\right)}{\color{blue}{x}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\sin x \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{6} \cdot y\right)\right), \color{blue}{x}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sin x \cdot \left(y \cdot y\right)\right), \left(\frac{1}{6} \cdot y\right)\right), x\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\sin x \cdot y\right) \cdot y\right), \left(\frac{1}{6} \cdot y\right)\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sin x \cdot y\right), y\right), \left(\frac{1}{6} \cdot y\right)\right), x\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, y\right), y\right), \left(\frac{1}{6} \cdot y\right)\right), x\right) \]

       8. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), y\right), y\right), \left(\frac{1}{6} \cdot y\right)\right), x\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), y\right), y\right), \left(y \cdot \frac{1}{6}\right)\right), x\right) \]

       10. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), y\right), y\right), \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right), x\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\left(\left(\sin x \cdot y\right) \cdot y\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10.2:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}} \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y \cdot \left(\sin x \cdot y\right)\right) \cdot \left(y \cdot 0.16666666666666666\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (* y (/ (sin x) x))
   (if (<= y 1.5e+49)
     (sinh y)
     (*
      (*
       y
       (+
        1.0
        (*
         (* y y)
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
      (+ 1.0 (* (* x x) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (sin(x) / x);
	} else if (y <= 1.5e+49) {
		tmp = sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * 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 <= 0.0065d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 1.5d+49) then
        tmp = sinh(y)
    else
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 1.5e+49) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = y * (math.sin(x) / x)
	elif y <= 1.5e+49:
		tmp = math.sinh(y)
	else:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 1.5e+49)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0065)
		tmp = y * (sin(x) / x);
	elseif (y <= 1.5e+49)
		tmp = sinh(y);
	else
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+49], N[Sinh[y], $MachinePrecision], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0064999999999999997

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 86.8%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{\sin x}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\sin x, \color{blue}{x}\right)\right) \]

      2. sin-lowering-sin.f64 67.9%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto y \cdot \color{blue}{\frac{\sin x}{x}} \]

   if 0.0064999999999999997 < y < 1.5000000000000001e49

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 50.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{x} \]

      2. div-inv N/A

         \[\leadsto \left(\sinh y \cdot \frac{1}{x}\right) \cdot x \]

      3. associate-*l* N/A

         \[\leadsto \sinh y \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 50.2%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\sinh y} \]

   if 1.5000000000000001e49 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}, x\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 72.5%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (/ x (/ x y))
   (if (<= y 2.7e+49)
     (sinh y)
     (*
      (*
       y
       (+
        1.0
        (*
         (* y y)
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
      (+ 1.0 (* (* x x) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = x / (x / y);
	} else if (y <= 2.7e+49) {
		tmp = sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * 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 <= 0.0065d0) then
        tmp = x / (x / y)
    else if (y <= 2.7d+49) then
        tmp = sinh(y)
    else
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = x / (x / y);
	} else if (y <= 2.7e+49) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = x / (x / y)
	elif y <= 2.7e+49:
		tmp = math.sinh(y)
	else:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 2.7e+49)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0065)
		tmp = x / (x / y);
	elseif (y <= 2.7e+49)
		tmp = sinh(y);
	else
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+49], N[Sinh[y], $MachinePrecision], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0064999999999999997

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 62.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 62.0%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 60.8%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 60.8%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 0.0064999999999999997 < y < 2.7000000000000001e49

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 50.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sinh y}{x} \cdot \color{blue}{x} \]

      2. div-inv N/A

         \[\leadsto \left(\sinh y \cdot \frac{1}{x}\right) \cdot x \]

      3. associate-*l* N/A

         \[\leadsto \sinh y \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      4. lft-mult-inverse N/A

         \[\leadsto \sinh y \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 50.2%

         \[\leadsto \mathsf{sinh.f64}\left(y\right) \]

   9. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\sinh y} \]

   if 2.7000000000000001e49 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}, x\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 72.5%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e+49)
   (* x (/ (sinh y) x))
   (*
    (*
     y
     (+
      1.0
      (*
       (* y y)
       (+
        0.16666666666666666
        (*
         y
         (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
    (+ 1.0 (* (* x x) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+49) {
		tmp = x * (sinh(y) / x);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * 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 <= 3.8d+49) then
        tmp = x * (sinh(y) / x)
    else
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+49) {
		tmp = x * (Math.sinh(y) / x);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e+49:
		tmp = x * (math.sinh(y) / x)
	else:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e+49)
		tmp = Float64(x * Float64(sinh(y) / x));
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e+49)
		tmp = x * (sinh(y) / x);
	else
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) * (1.0 + ((x * x) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e+49], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * 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] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999999e49

   1. Initial program 85.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 74.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   if 3.7999999999999999e49 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}, x\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot y\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \sin x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}}{x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 72.5%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(y \cdot \left(\frac{1}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0065)
   (/ x (/ x y))
   (*
    (* x (+ 1.0 (* (* x x) -0.16666666666666666)))
    (* y (* (/ 1.0 x) (+ 1.0 (* (* y y) 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = x / (x / y);
	} else {
		tmp = (x * (1.0 + ((x * x) * -0.16666666666666666))) * (y * ((1.0 / x) * (1.0 + ((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 <= 0.0065d0) then
        tmp = x / (x / y)
    else
        tmp = (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))) * (y * ((1.0d0 / x) * (1.0d0 + ((y * y) * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0065) {
		tmp = x / (x / y);
	} else {
		tmp = (x * (1.0 + ((x * x) * -0.16666666666666666))) * (y * ((1.0 / x) * (1.0 + ((y * y) * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0065:
		tmp = x / (x / y)
	else:
		tmp = (x * (1.0 + ((x * x) * -0.16666666666666666))) * (y * ((1.0 / x) * (1.0 + ((y * y) * 0.16666666666666666))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0065)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))) * Float64(y * Float64(Float64(1.0 / x) * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0065)
		tmp = x / (x / y);
	else
		tmp = (x * (1.0 + ((x * x) * -0.16666666666666666))) * (y * ((1.0 / x) * (1.0 + ((y * y) * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0065], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0064999999999999997

   1. Initial program 84.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 62.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 62.0%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 60.8%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 60.8%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 0.0064999999999999997 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{\frac{1}{6} \cdot {y}^{2}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{{y}^{2} \cdot \frac{1}{6}}{x} + \frac{1}{x}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6}}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left({y}^{2} \cdot \frac{\frac{1}{6} \cdot 1}{x} + \frac{1}{x}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{1}{x} + \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{x} + \frac{1}{x}\right)\right)\right) \]

      9. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{1}{x}}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{1}}{x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right), \left(\frac{\color{blue}{1}}{x}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(\frac{1}{x}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 59.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right)\right) \]

   7. Simplified 59.8%

      \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{x}\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{y}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

      6. *-lowering-*.f64 66.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]

   10. Simplified 66.5%

       \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \left(y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{x}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0065:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(y \cdot \left(\frac{1}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.008333333333333333\\ \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + t\_0\right)}{x}\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{t\_0}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 0.008333333333333333)))
   (if (<= x 1.52e+19)
     (* x (* y (/ (+ 1.0 (* (* y y) (+ 0.16666666666666666 t_0))) x)))
     (if (<= x 1.7e+167)
       (*
        (* y (* y y))
        (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))
       (* x (* y (* (* y y) (/ t_0 x))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 0.008333333333333333;
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * (y * ((1.0 + ((y * y) * (0.16666666666666666 + t_0))) / x));
	} else if (x <= 1.7e+167) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = x * (y * ((y * y) * (t_0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * 0.008333333333333333d0
    if (x <= 1.52d+19) then
        tmp = x * (y * ((1.0d0 + ((y * y) * (0.16666666666666666d0 + t_0))) / x))
    else if (x <= 1.7d+167) then
        tmp = (y * (y * y)) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0)))
    else
        tmp = x * (y * ((y * y) * (t_0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 0.008333333333333333;
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * (y * ((1.0 + ((y * y) * (0.16666666666666666 + t_0))) / x));
	} else if (x <= 1.7e+167) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = x * (y * ((y * y) * (t_0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 0.008333333333333333
	tmp = 0
	if x <= 1.52e+19:
		tmp = x * (y * ((1.0 + ((y * y) * (0.16666666666666666 + t_0))) / x))
	elif x <= 1.7e+167:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	else:
		tmp = x * (y * ((y * y) * (t_0 / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 0.008333333333333333)
	tmp = 0.0
	if (x <= 1.52e+19)
		tmp = Float64(x * Float64(y * Float64(Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + t_0))) / x)));
	elseif (x <= 1.7e+167)
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776)));
	else
		tmp = Float64(x * Float64(y * Float64(Float64(y * y) * Float64(t_0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 0.008333333333333333;
	tmp = 0.0;
	if (x <= 1.52e+19)
		tmp = x * (y * ((1.0 + ((y * y) * (0.16666666666666666 + t_0))) / x));
	elseif (x <= 1.7e+167)
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	else
		tmp = x * (y * ((y * y) * (t_0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]}, If[LessEqual[x, 1.52e+19], N[(x * N[(y * N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e+167], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.52e19

   1. Initial program 84.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {y}^{2}}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{{y}^{2}}{x} \cdot \frac{1}{120}\right) \cdot {y}^{2} + \left(\color{blue}{\frac{1}{6}} \cdot \frac{1}{x}\right) \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right)} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot 1}{x} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6}}{x} \cdot {y}^{2}\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{{y}^{2}}{x} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\frac{{y}^{2}}{x}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({y}^{2}\right), x\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   10. Simplified 73.4%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{x} + \frac{y \cdot y}{x} \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}{x}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right), \color{blue}{x}\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right), x\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right), x\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right), x\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right), x\right)\right)\right) \]

       10. *-lowering-*.f64 73.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right), x\right)\right)\right) \]

   13. Simplified 73.4%

       \[\leadsto x \cdot \left(y \cdot \color{blue}{\frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)}{x}}\right) \]

   if 1.52e19 < x < 1.7e167

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 38.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 38.7%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{3}} \]

       3. distribute-rgt-out N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       14. *-lowering-*.f64 30.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   13. Simplified 30.9%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]

   if 1.7e167 < x

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 67.8%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {y}^{2}}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{{y}^{2}}{x} \cdot \frac{1}{120}\right) \cdot {y}^{2} + \left(\color{blue}{\frac{1}{6}} \cdot \frac{1}{x}\right) \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right)} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot 1}{x} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6}}{x} \cdot {y}^{2}\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{{y}^{2}}{x} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\frac{{y}^{2}}{x}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({y}^{2}\right), x\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{x} + \frac{y \cdot y}{x} \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{4}}{x}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{4}}{\color{blue}{x}}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}}{x}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)}{x}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}}{x}\right)\right)\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{2}}{x} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} \cdot \frac{{y}^{2}}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} \cdot \frac{{y}^{2}}{x}\right)\right)\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{\frac{1}{120} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\left(\frac{1}{120} \cdot {y}^{2}\right), \color{blue}{x}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\left({y}^{2} \cdot \frac{1}{120}\right), x\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

       16. *-lowering-*.f64 68.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

   13. Simplified 68.4%

       \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{\left(y \cdot y\right) \cdot 0.008333333333333333}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{\left(y \cdot y\right) \cdot 0.008333333333333333}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.52e+19)
   (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
   (if (<= x 1.25e+169)
     (*
      (* y (* y y))
      (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))
     (* x (* y (* (* y y) (/ (* (* y y) 0.008333333333333333) x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = x * (y * ((y * y) * (((y * y) * 0.008333333333333333) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.52d+19) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else if (x <= 1.25d+169) then
        tmp = (y * (y * y)) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0)))
    else
        tmp = x * (y * ((y * y) * (((y * y) * 0.008333333333333333d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = x * (y * ((y * y) * (((y * y) * 0.008333333333333333) / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.52e+19:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	elif x <= 1.25e+169:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	else:
		tmp = x * (y * ((y * y) * (((y * y) * 0.008333333333333333) / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.52e+19)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	elseif (x <= 1.25e+169)
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776)));
	else
		tmp = Float64(x * Float64(y * Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * 0.008333333333333333) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.52e+19)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	elseif (x <= 1.25e+169)
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	else
		tmp = x * (y * ((y * y) * (((y * y) * 0.008333333333333333) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.52e+19], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+169], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.52e19

   1. Initial program 84.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 72.2%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

   if 1.52e19 < x < 1.25000000000000004e169

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 38.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 38.7%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{3}} \]

       3. distribute-rgt-out N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       14. *-lowering-*.f64 30.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   13. Simplified 30.9%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]

   if 1.25000000000000004e169 < x

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 67.8%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right) + \frac{1}{x}\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{x} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot \frac{{y}^{2}}{x} + \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot {y}^{2}}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\left(\frac{{y}^{2}}{x} \cdot \frac{1}{120}\right) \cdot {y}^{2} + \left(\color{blue}{\frac{1}{6}} \cdot \frac{1}{x}\right) \cdot {y}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{x}\right)} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot 1}{x} \cdot {\color{blue}{y}}^{2}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6}}{x} \cdot {y}^{2}\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{\frac{1}{6} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\frac{{y}^{2}}{x}}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right) + \frac{{y}^{2}}{x} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      13. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)}\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{y}^{2}}{x} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\left(\frac{{y}^{2}}{x}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({y}^{2}\right), x\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right), \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(\frac{1}{x} + \frac{y \cdot y}{x} \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{4}}{x}\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{4}}{\color{blue}{x}}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{\left(2 \cdot 2\right)}}{x}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot \left({y}^{2} \cdot {y}^{2}\right)}{x}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}}{x}\right)\right)\right) \]

       5. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{\frac{1}{120} \cdot {y}^{2}}{x} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right) \cdot {\color{blue}{y}}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot \frac{{y}^{2}}{x}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} \cdot \frac{{y}^{2}}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} \cdot \frac{{y}^{2}}{x}\right)\right)\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{\frac{1}{120} \cdot {y}^{2}}{\color{blue}{x}}\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\left(\frac{1}{120} \cdot {y}^{2}\right), \color{blue}{x}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\left({y}^{2} \cdot \frac{1}{120}\right), x\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

       16. *-lowering-*.f64 68.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right), x\right)\right)\right)\right) \]

   13. Simplified 68.4%

       \[\leadsto x \cdot \left(y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{\left(y \cdot y\right) \cdot 0.008333333333333333}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot \frac{\left(y \cdot y\right) \cdot 0.008333333333333333}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3)
   (/ x (/ x y))
   (*
    (* y 0.16666666666666666)
    (* (* x (+ 1.0 (* (* x x) -0.16666666666666666))) (/ (* y y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3) {
		tmp = x / (x / y);
	} else {
		tmp = (y * 0.16666666666666666) * ((x * (1.0 + ((x * x) * -0.16666666666666666))) * ((y * y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.3d0) then
        tmp = x / (x / y)
    else
        tmp = (y * 0.16666666666666666d0) * ((x * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))) * ((y * y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3) {
		tmp = x / (x / y);
	} else {
		tmp = (y * 0.16666666666666666) * ((x * (1.0 + ((x * x) * -0.16666666666666666))) * ((y * y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3:
		tmp = x / (x / y)
	else:
		tmp = (y * 0.16666666666666666) * ((x * (1.0 + ((x * x) * -0.16666666666666666))) * ((y * y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(Float64(y * 0.16666666666666666) * Float64(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))) * Float64(Float64(y * y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3)
		tmp = x / (x / y);
	else
		tmp = (y * 0.16666666666666666) * ((x * (1.0 + ((x * x) * -0.16666666666666666))) * ((y * y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999998

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 61.7%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 60.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 3.2999999999999998 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 53.8%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 58.3%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

       6. *-lowering-*.f64 66.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]

   13. Simplified 66.1%

       \[\leadsto \left(\color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.52e+19)
   (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
   (if (<= x 1.25e+169)
     (*
      (* y (* y y))
      (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))
     (* (* y 0.16666666666666666) (* x (/ (* y y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.52d+19) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else if (x <= 1.25d+169) then
        tmp = (y * (y * y)) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0)))
    else
        tmp = (y * 0.16666666666666666d0) * (x * ((y * y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.52e+19:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	elif x <= 1.25e+169:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	else:
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.52e+19)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	elseif (x <= 1.25e+169)
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776)));
	else
		tmp = Float64(Float64(y * 0.16666666666666666) * Float64(x * Float64(Float64(y * y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.52e+19)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	elseif (x <= 1.25e+169)
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	else
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.52e+19], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+169], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.52e19

   1. Initial program 84.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 72.2%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

   if 1.52e19 < x < 1.25000000000000004e169

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 38.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 38.7%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right) + \frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \color{blue}{\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{3}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{1}{6} \cdot {y}^{3} + \left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{3}} \]

       3. distribute-rgt-out N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       14. *-lowering-*.f64 30.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   13. Simplified 30.9%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]

   if 1.25000000000000004e169 < x

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 67.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 63.3%

       \[\leadsto \left(\color{blue}{x} \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.52e+19)
   (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
   (if (<= x 1.25e+169)
     (*
      y
      (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))
     (* (* y 0.16666666666666666) (* x (/ (* y y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.52d+19) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else if (x <= 1.25d+169) then
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    else
        tmp = (y * 0.16666666666666666d0) * (x * ((y * y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.52e+19) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (x <= 1.25e+169) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.52e+19:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	elif x <= 1.25e+169:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	else:
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.52e+19)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	elseif (x <= 1.25e+169)
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	else
		tmp = Float64(Float64(y * 0.16666666666666666) * Float64(x * Float64(Float64(y * y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.52e+19)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	elseif (x <= 1.25e+169)
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	else
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.52e+19], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+169], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.52e19

   1. Initial program 84.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 72.2%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

   if 1.52e19 < x < 1.25000000000000004e169

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 22.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 22.9%

       \[\leadsto y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)}\right) \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

       4. cube-mult N/A

          \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto y \cdot \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto y \cdot \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)}\right) \]

   13. Simplified 22.9%

       \[\leadsto \color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right) \cdot 0.16666666666666666\right)\right)\right)} \]

   14. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right) + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right) \]

       2. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\left(\frac{1}{6} \cdot \frac{-1}{6}\right) \cdot {x}^{2} + \frac{1}{6}\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6}\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{1}\right)\right)\right) \]

       6. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       11. distribute-lft-in N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot 1 + \color{blue}{\frac{1}{6} \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \color{blue}{\frac{1}{6}} \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \left(\frac{1}{6} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 30.9%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   16. Simplified 30.9%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]

   if 1.25000000000000004e169 < x

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 67.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 63.3%

       \[\leadsto \left(\color{blue}{x} \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10.0)
   (/ x (/ x y))
   (if (<= y 1.35e+79)
     (* y (+ 1.0 (* (* x x) -0.16666666666666666)))
     (* (* y 0.16666666666666666) (* x (/ (* y y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10.0) {
		tmp = x / (x / y);
	} else if (y <= 1.35e+79) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 10.0d0) then
        tmp = x / (x / y)
    else if (y <= 1.35d+79) then
        tmp = y * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = (y * 0.16666666666666666d0) * (x * ((y * y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 10.0) {
		tmp = x / (x / y);
	} else if (y <= 1.35e+79) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 10.0:
		tmp = x / (x / y)
	elif y <= 1.35e+79:
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10.0)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 1.35e+79)
		tmp = Float64(y * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(y * 0.16666666666666666) * Float64(x * Float64(Float64(y * y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 10.0)
		tmp = x / (x / y);
	elseif (y <= 1.35e+79)
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = (y * 0.16666666666666666) * (x * ((y * y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10.0], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+79], N[(y * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.16666666666666666), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 10

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 61.7%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 60.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 10 < y < 1.35e79

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 3.2%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 48.6%

       \[\leadsto y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) + y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 39.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)} \]

   14. Taylor expanded in y around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \color{blue}{y}\right) \]
   15. Step-by-step derivation

   16. Simplified 35.0%

       \[\leadsto \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \color{blue}{y} \]

   if 1.35e79 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 78.5%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 80.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 80.7%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, y\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 58.4%

       \[\leadsto \left(\color{blue}{x} \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot 0.16666666666666666\right) \cdot \left(x \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4)
   (/ x (/ x y))
   (if (<= y 8.6e+104)
     (* y (+ 1.0 (* (* x x) -0.16666666666666666)))
     (* y (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4) {
		tmp = x / (x / y);
	} else if (y <= 8.6e+104) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.4d0) then
        tmp = x / (x / y)
    else if (y <= 8.6d+104) then
        tmp = y * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = y * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4) {
		tmp = x / (x / y);
	} else if (y <= 8.6e+104) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4:
		tmp = x / (x / y)
	elif y <= 8.6e+104:
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = y * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 8.6e+104)
		tmp = Float64(y * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4)
		tmp = x / (x / y);
	elseif (y <= 8.6e+104)
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = y * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+104], N[(y * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4000000000000004

   1. Initial program 84.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 61.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 61.7%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

       4. /-lowering-/.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 60.5%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 4.4000000000000004 < y < 8.6000000000000003e104

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 4.0%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 51.5%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 51.5%

       \[\leadsto y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)}\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) + y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) + \color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 44.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 44.4%

       \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)} \]

   14. Taylor expanded in y around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), \color{blue}{y}\right) \]
   15. Step-by-step derivation

   16. Simplified 36.4%

       \[\leadsto \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \color{blue}{y} \]

   if 8.6000000000000003e104 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 87.9%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 87.9%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left({y}^{2} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       8. *-lowering-*.f64 63.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 63.2%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 66.3% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x)))

C

double code(double x, double y) {
	return x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
end function

Java

public static double code(double x, double y) {
	return x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
}

Python

def code(x, y):
	return x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)

Julia

function code(x, y)
	return Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
end

Wolfram

code[x_, y_] := N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 71.3%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}, x\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), x\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), x\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left({y}^{2}\right)\right)\right)\right), x\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot y\right)\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 65.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

10. Simplified 65.7%

    \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

11. Final simplification 65.7%

    \[\leadsto x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x} \]
12. Add Preprocessing

Alternative 22: 57.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e+99)
   (* x (/ 1.0 (/ x y)))
   (* y (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+99) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d+99) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e+99) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e+99:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e+99)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e+99)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e+99], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.19999999999999999e99

   1. Initial program 86.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 73.3%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 56.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 56.2%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 56.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 56.3%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]

   if 3.19999999999999999e99 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x} + \frac{\sin x}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot \frac{{y}^{2} \cdot \sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\frac{\sin x}{x}}\right), y \cdot \frac{\sin x}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{\sin x}{x}}, y \cdot \frac{\sin x}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\color{blue}{\sin x}}{x}, y \cdot \frac{\sin x}{x}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(y, {y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)}, y \cdot \frac{\sin x}{x}\right) \]

      7. fma-define N/A

         \[\leadsto y \cdot \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) + \color{blue}{y \cdot \frac{\sin x}{x}} \]

      8. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) + \frac{\sin x}{x}\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \frac{\sin x}{x} + \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\sin x}{x} + \frac{\sin \color{blue}{x}}{x}\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\frac{\sin x}{x}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{\color{blue}{\sin x}}{x}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right), \color{blue}{\left(\frac{\sin x}{x}\right)}\right)\right) \]

   7. Simplified 82.1%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{y}^{3} \cdot \sin x}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{y}^{3} \cdot \sin x}{x} \cdot \color{blue}{\frac{1}{6}} \]

      2. associate-/l* N/A

         \[\leadsto \left({y}^{3} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6} \]

      3. associate-*r* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)} \]

      4. *-commutative N/A

         \[\leadsto {y}^{3} \cdot \left(\frac{1}{6} \cdot \color{blue}{\frac{\sin x}{x}}\right) \]

      5. cube-mult N/A

         \[\leadsto \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot \frac{\sin x}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(y \cdot {y}^{2}\right) \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right) \]

      7. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \frac{\sin x}{x}\right)\right) \cdot \color{blue}{y} \]

      9. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \left(\frac{\sin x}{x} \cdot \frac{1}{6}\right)\right) \cdot y \]

      10. associate-*r* N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{\sin x}{x}\right) \cdot \frac{1}{6}\right) \cdot y \]

      11. associate-/l* N/A

          \[\leadsto \left(\frac{{y}^{2} \cdot \sin x}{x} \cdot \frac{1}{6}\right) \cdot y \]

      12. associate-*l* N/A

          \[\leadsto \frac{{y}^{2} \cdot \sin x}{x} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{{y}^{2} \cdot \sin x}{x}\right), \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin x \cdot {y}^{2}}{x}\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sin x \cdot \frac{{y}^{2}}{x}\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{{y}^{2}}{x}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right) \]

      17. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{{y}^{2}}{x}\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left({y}^{2}\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\left(y \cdot y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \left(\frac{1}{6} \cdot y\right)\right) \]

      21. *-lowering-*.f64 82.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, y\right), x\right)\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{y}\right)\right) \]

   10. Simplified 82.1%

       \[\leadsto \color{blue}{\left(\sin x \cdot \frac{y \cdot y}{x}\right) \cdot \left(0.16666666666666666 \cdot y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   12. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left({y}^{2} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       8. *-lowering-*.f64 58.8%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Simplified 58.8%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 51.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ 1.0 (/ x y))))

C

double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 / (x / y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Python

def code(x, y):
	return x * (1.0 / (x / y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 / Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 / (x / y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 71.3%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 50.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 50.8%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

    3. /-lowering-/.f64 50.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

12. Applied egg-rr 50.8%

    \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 24: 50.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ y x)))

C

double code(double x, double y) {
	return x * (y / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (y / x)
end function

Java

public static double code(double x, double y) {
	return x * (y / x);
}

Python

def code(x, y):
	return x * (y / x)

Julia

function code(x, y)
	return Float64(x * Float64(y / x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (y / x);
end

Wolfram

code[x_, y_] := N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 71.3%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 50.8%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 50.8%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 25: 27.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 88.5%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

   3. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

   5. sinh-lowering-sinh.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
6. Step-by-step derivation

7. Simplified 71.3%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
9. Step-by-step derivation

10. Simplified 29.5%

    \[\leadsto \color{blue}{y} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))