Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.9%

Time: 14.6s

Alternatives: 23

Speedup: 1.0×

• 49.3% 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: 88.9% 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 87.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ t_1 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(t\_0 \cdot t\_1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_1 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t\_0 \cdot \frac{\sin x}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y))))
        (t_1 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 1.35e-8)
     (/ y (/ x (sin x)))
     (if (<= y 2.6e+48)
       (sinh y)
       (if (<= y 4.8e+124)
         (*
          y
          (+
           (* t_0 t_1)
           (*
            (* y y)
            (*
             y
             (*
              y
              (*
               t_1
               (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
         (* y (* t_0 (/ (sin x) x))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y / (x / sin(x));
	} else if (y <= 2.6e+48) {
		tmp = sinh(y);
	} else if (y <= 4.8e+124) {
		tmp = y * ((t_0 * t_1) + ((y * y) * (y * (y * (t_1 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	} else {
		tmp = y * (t_0 * (sin(x) / 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    t_1 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 1.35d-8) then
        tmp = y / (x / sin(x))
    else if (y <= 2.6d+48) then
        tmp = sinh(y)
    else if (y <= 4.8d+124) then
        tmp = y * ((t_0 * t_1) + ((y * y) * (y * (y * (t_1 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    else
        tmp = y * (t_0 * (sin(x) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double t_1 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 2.6e+48) {
		tmp = Math.sinh(y);
	} else if (y <= 4.8e+124) {
		tmp = y * ((t_0 * t_1) + ((y * y) * (y * (y * (t_1 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	} else {
		tmp = y * (t_0 * (Math.sin(x) / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	t_1 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 1.35e-8:
		tmp = y / (x / math.sin(x))
	elif y <= 2.6e+48:
		tmp = math.sinh(y)
	elif y <= 4.8e+124:
		tmp = y * ((t_0 * t_1) + ((y * y) * (y * (y * (t_1 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	else:
		tmp = y * (t_0 * (math.sin(x) / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	t_1 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 1.35e-8)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 2.6e+48)
		tmp = sinh(y);
	elseif (y <= 4.8e+124)
		tmp = Float64(y * Float64(Float64(t_0 * t_1) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_1 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	else
		tmp = Float64(y * Float64(t_0 * Float64(sin(x) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	t_1 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 1.35e-8)
		tmp = y / (x / sin(x));
	elseif (y <= 2.6e+48)
		tmp = sinh(y);
	elseif (y <= 4.8e+124)
		tmp = y * ((t_0 * t_1) + ((y * y) * (y * (y * (t_1 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	else
		tmp = y * (t_0 * (sin(x) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-8], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+48], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 4.8e+124], N[(y * N[(N[(t$95$0 * t$95$1), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$1 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.35000000000000001e-8

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

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

      11. sin-lowering-sin.f64 72.6%

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

   7. Applied egg-rr 72.6%

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

   if 1.35000000000000001e-8 < y < 2.59999999999999995e48

   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 100.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2.59999999999999995e48 < y < 4.80000000000000013e124

   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 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.0%

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

   if 4.80000000000000013e124 < 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 100.0%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-8)
   (/ y (/ x (sin x)))
   (if (<= y 8.2e+124)
     (/ (* (sinh y) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))) x)
     (* y (* (+ 1.0 (* 0.16666666666666666 (* y y))) (/ (sin x) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-8) {
		tmp = y / (x / sin(x));
	} else if (y <= 8.2e+124) {
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d-8) then
        tmp = y / (x / sin(x))
    else if (y <= 8.2d+124) then
        tmp = (sinh(y) * (x * (1.0d0 + ((x * x) * (-0.16666666666666666d0))))) / x
    else
        tmp = y * ((1.0d0 + (0.16666666666666666d0 * (y * y))) * (sin(x) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e-8) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 8.2e+124) {
		tmp = (Math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	} else {
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (Math.sin(x) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-8:
		tmp = y / (x / math.sin(x))
	elif y <= 8.2e+124:
		tmp = (math.sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x
	else:
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (math.sin(x) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-8)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 8.2e+124)
		tmp = Float64(Float64(sinh(y) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)))) / x);
	else
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(sin(x) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e-8)
		tmp = y / (x / sin(x));
	elseif (y <= 8.2e+124)
		tmp = (sinh(y) * (x * (1.0 + ((x * x) * -0.16666666666666666)))) / x;
	else
		tmp = y * ((1.0 + (0.16666666666666666 * (y * y))) * (sin(x) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-8], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+124], 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[(y * N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.0000000000000002e-8

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

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

      11. sin-lowering-sin.f64 72.6%

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

   7. Applied egg-rr 72.6%

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

   if 8.0000000000000002e-8 < y < 8.20000000000000002e124

   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 85.7%

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

   5. Simplified 85.7%

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

   if 8.20000000000000002e124 < 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 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sinh y \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{\sin x}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+49}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 1.35e-8)
     (/ y (/ x (sin x)))
     (if (<= y 1.6e+49)
       (sinh y)
       (*
        y
        (+
         (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
         (*
          (* y y)
          (*
           y
           (*
            y
            (*
             t_0
             (+
              0.008333333333333333
              (* (* y y) 0.0001984126984126984))))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y / (x / sin(x));
	} else if (y <= 1.6e+49) {
		tmp = sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 1.35d-8) then
        tmp = y / (x / sin(x))
    else if (y <= 1.6d+49) then
        tmp = sinh(y)
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 1.6e+49) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 1.35e-8:
		tmp = y / (x / math.sin(x))
	elif y <= 1.6e+49:
		tmp = math.sinh(y)
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 1.35e-8)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 1.6e+49)
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 1.35e-8)
		tmp = y / (x / sin(x));
	elseif (y <= 1.6e+49)
		tmp = sinh(y);
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-8], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+49], N[Sinh[y], $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.35000000000000001e-8

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. clear-num N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

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

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

      11. sin-lowering-sin.f64 72.6%

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

   7. Applied egg-rr 72.6%

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

   if 1.35000000000000001e-8 < y < 1.60000000000000007e49

   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 100.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.60000000000000007e49 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

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

Alternative 5: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 1.35e-8)
     (* y (/ (sin x) x))
     (if (<= y 5e+44)
       (sinh y)
       (*
        y
        (+
         (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
         (*
          (* y y)
          (*
           y
           (*
            y
            (*
             t_0
             (+
              0.008333333333333333
              (* (* y y) 0.0001984126984126984))))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y * (sin(x) / x);
	} else if (y <= 5e+44) {
		tmp = sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 1.35d-8) then
        tmp = y * (sin(x) / x)
    else if (y <= 5d+44) then
        tmp = sinh(y)
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 5e+44) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 1.35e-8:
		tmp = y * (math.sin(x) / x)
	elif y <= 5e+44:
		tmp = math.sinh(y)
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 1.35e-8)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 5e+44)
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 1.35e-8)
		tmp = y * (sin(x) / x);
	elseif (y <= 5e+44)
		tmp = sinh(y);
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-8], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+44], N[Sinh[y], $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.35000000000000001e-8

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. sin-lowering-sin.f64 72.5%

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

   7. Applied egg-rr 72.5%

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

   if 1.35000000000000001e-8 < y < 4.9999999999999996e44

   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 100.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 4.9999999999999996e44 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 10^{+48}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 1.35e-8)
     (* (sin x) (/ y x))
     (if (<= y 1e+48)
       (sinh y)
       (*
        y
        (+
         (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
         (*
          (* y y)
          (*
           y
           (*
            y
            (*
             t_0
             (+
              0.008333333333333333
              (* (* y y) 0.0001984126984126984))))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = sin(x) * (y / x);
	} else if (y <= 1e+48) {
		tmp = sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 1.35d-8) then
        tmp = sin(x) * (y / x)
    else if (y <= 1d+48) then
        tmp = sinh(y)
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 1.35e-8) {
		tmp = Math.sin(x) * (y / x);
	} else if (y <= 1e+48) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 1.35e-8:
		tmp = math.sin(x) * (y / x)
	elif y <= 1e+48:
		tmp = math.sinh(y)
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 1.35e-8)
		tmp = Float64(sin(x) * Float64(y / x));
	elseif (y <= 1e+48)
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 1.35e-8)
		tmp = sin(x) * (y / x);
	elseif (y <= 1e+48)
		tmp = sinh(y);
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-8], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+48], N[Sinh[y], $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.35000000000000001e-8

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 79.8%

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

   7. Simplified 79.8%

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

   if 1.35000000000000001e-8 < y < 1.00000000000000004e48

   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 100.0%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.00000000000000004e48 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

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

Alternative 7: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 4 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 10^{+48}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 4e-52)
     (/ x (/ x y))
     (if (<= y 1e+48)
       (sinh y)
       (*
        y
        (+
         (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
         (*
          (* y y)
          (*
           y
           (*
            y
            (*
             t_0
             (+
              0.008333333333333333
              (* (* y y) 0.0001984126984126984))))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 4e-52) {
		tmp = x / (x / y);
	} else if (y <= 1e+48) {
		tmp = sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 4d-52) then
        tmp = x / (x / y)
    else if (y <= 1d+48) then
        tmp = sinh(y)
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 4e-52) {
		tmp = x / (x / y);
	} else if (y <= 1e+48) {
		tmp = Math.sinh(y);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 4e-52:
		tmp = x / (x / y)
	elif y <= 1e+48:
		tmp = math.sinh(y)
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 4e-52)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 1e+48)
		tmp = sinh(y);
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 4e-52)
		tmp = x / (x / y);
	elseif (y <= 1e+48)
		tmp = sinh(y);
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4e-52], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+48], N[Sinh[y], $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4e-52

   1. Initial program 82.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.1%

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

   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 56.7%

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

   10. Simplified 56.7%

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

       1. clear-num N/A

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

       2. 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 58.0%

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

   12. Applied egg-rr 58.0%

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

   if 4e-52 < y < 1.00000000000000004e48

   1. Initial program 93.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 69.4%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

         \[\leadsto \sinh y \]

      6. sinh-lowering-sinh.f64 69.4%

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

   9. Applied egg-rr 69.4%

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

   if 1.00000000000000004e48 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

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

Alternative 8: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 3.6 \cdot 10^{+44}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 3.6e+44)
     (* x (/ (sinh y) x))
     (*
      y
      (+
       (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
       (*
        (* y y)
        (*
         y
         (*
          y
          (*
           t_0
           (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 3.6e+44) {
		tmp = x * (sinh(y) / x);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 3.6d+44) then
        tmp = x * (sinh(y) / x)
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 3.6e+44) {
		tmp = x * (Math.sinh(y) / x);
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 3.6e+44:
		tmp = x * (math.sinh(y) / x)
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 3.6e+44)
		tmp = Float64(x * Float64(sinh(y) / x));
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 3.6e+44)
		tmp = x * (sinh(y) / x);
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.6e+44], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6e44

   1. Initial program 83.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.2%

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

   if 3.6e44 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{if}\;y \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t\_0 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(t\_0 \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x -0.16666666666666666)))))
   (if (<= y 5e+44)
     (/
      x
      (/
       (/ x y)
       (+
        1.0
        (*
         y
         (*
          y
          (+
           0.16666666666666666
           (*
            y
            (*
             y
             (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))))))
     (*
      y
      (+
       (* (+ 1.0 (* 0.16666666666666666 (* y y))) t_0)
       (*
        (* y y)
        (*
         y
         (*
          y
          (*
           t_0
           (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 5e+44) {
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (-0.16666666666666666d0)))
    if (y <= 5d+44) then
        tmp = x / ((x / y) / (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))))))
    else
        tmp = y * (((1.0d0 + (0.16666666666666666d0 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * -0.16666666666666666));
	double tmp;
	if (y <= 5e+44) {
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	} else {
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * -0.16666666666666666))
	tmp = 0
	if y <= 5e+44:
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))))
	else:
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))
	tmp = 0.0
	if (y <= 5e+44)
		tmp = Float64(x / Float64(Float64(x / y) / Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984)))))))))));
	else
		tmp = Float64(y * Float64(Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * t_0) + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(t_0 * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * -0.16666666666666666));
	tmp = 0.0;
	if (y <= 5e+44)
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	else
		tmp = y * (((1.0 + (0.16666666666666666 * (y * y))) * t_0) + ((y * y) * (y * (y * (t_0 * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5e+44], N[(x / N[(N[(x / y), $MachinePrecision] / N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(t$95$0 * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999996e44

   1. Initial program 83.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.2%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.2%

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

   10. Simplified 66.2%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

       4. associate-/r* N/A

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

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

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

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

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

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

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. associate-*l* N/A

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

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

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

   12. Applied egg-rr 66.9%

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

   if 4.9999999999999996e44 < y

   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 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 80.3%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e+114)
   (/
    x
    (/
     (/ x y)
     (+
      1.0
      (*
       y
       (*
        y
        (+
         0.16666666666666666
         (*
          y
          (*
           y
           (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))))))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+114) {
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d+114) then
        tmp = x / ((x / y) / (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))))))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+114) {
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e+114:
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e+114)
		tmp = Float64(x / Float64(Float64(x / y) / Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984)))))))))));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e+114)
		tmp = x / ((x / y) / (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))))));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e+114], N[(x / N[(N[(x / y), $MachinePrecision] / N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e114

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.6%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.7%

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

   10. Simplified 66.7%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

       4. associate-/r* N/A

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

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

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

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

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

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

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. associate-*l* N/A

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

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

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

   12. Applied egg-rr 67.4%

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

   if 2.1e114 < y

   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 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 81.6%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.65e+48)
   (*
    x
    (/
     (*
      y
      (+
       1.0
       (*
        (* y y)
        (+
         0.16666666666666666
         (*
          (* y y)
          (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
     x))
   (if (<= x 5.5e+126)
     (*
      y
      (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))
     (* x (/ (* y (* 0.008333333333333333 (* y (* y (* y y))))) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.65e+48) {
		tmp = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) / x);
	} else if (x <= 5.5e+126) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (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 <= 1.65d+48) then
        tmp = x * ((y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))) / x)
    else if (x <= 5.5d+126) then
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    else
        tmp = x * ((y * (0.008333333333333333d0 * (y * (y * (y * y))))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.65e+48) {
		tmp = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) / x);
	} else if (x <= 5.5e+126) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.65e+48:
		tmp = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) / x)
	elif x <= 5.5e+126:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	else:
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.65e+48)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))) / x));
	elseif (x <= 5.5e+126)
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	else
		tmp = Float64(x * Float64(Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * Float64(y * y))))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.65e+48)
		tmp = x * ((y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) / x);
	elseif (x <= 5.5e+126)
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	else
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.65e+48], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+126], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(0.008333333333333333 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.65000000000000011e48

   1. Initial program 84.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.2%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 74.4%

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

   10. Simplified 74.4%

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

   if 1.65000000000000011e48 < x < 5.5000000000000004e126

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 40.3%

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

   5. Simplified 40.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 40.4%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 42.9%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 42.9%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]

   if 5.5000000000000004e126 < x

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 51.0%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left({y}^{4}\right)\right)\right), x\right)\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left({y}^{3}\right)\right)\right)\right), x\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 55.0%

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

   13. Simplified 55.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right) \cdot \frac{y}{x}\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.7 \cdot 10^{+239}:\\ \;\;\;\;y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* (+ 1.0 (* x (* x -0.16666666666666666))) (/ y x)))))
   (if (<= y 1.8e-14)
     (/ x (/ x y))
     (if (<= y 4e+92)
       t_0
       (if (<= y 7.7e+239)
         (* y (+ 1.0 (* 0.16666666666666666 (* y y))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = x * ((1.0 + (x * (x * -0.16666666666666666))) * (y / x));
	double tmp;
	if (y <= 1.8e-14) {
		tmp = x / (x / y);
	} else if (y <= 4e+92) {
		tmp = t_0;
	} else if (y <= 7.7e+239) {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((1.0d0 + (x * (x * (-0.16666666666666666d0)))) * (y / x))
    if (y <= 1.8d-14) then
        tmp = x / (x / y)
    else if (y <= 4d+92) then
        tmp = t_0
    else if (y <= 7.7d+239) then
        tmp = y * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((1.0 + (x * (x * -0.16666666666666666))) * (y / x));
	double tmp;
	if (y <= 1.8e-14) {
		tmp = x / (x / y);
	} else if (y <= 4e+92) {
		tmp = t_0;
	} else if (y <= 7.7e+239) {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((1.0 + (x * (x * -0.16666666666666666))) * (y / x))
	tmp = 0
	if y <= 1.8e-14:
		tmp = x / (x / y)
	elif y <= 4e+92:
		tmp = t_0
	elif y <= 7.7e+239:
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))) * Float64(y / x)))
	tmp = 0.0
	if (y <= 1.8e-14)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 4e+92)
		tmp = t_0;
	elseif (y <= 7.7e+239)
		tmp = Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((1.0 + (x * (x * -0.16666666666666666))) * (y / x));
	tmp = 0.0;
	if (y <= 1.8e-14)
		tmp = x / (x / y);
	elseif (y <= 4e+92)
		tmp = t_0;
	elseif (y <= 7.7e+239)
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e-14], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+92], t$95$0, If[LessEqual[y, 7.7e+239], N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7999999999999999e-14

   1. Initial program 82.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 67.6%

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

   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 56.4%

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

   10. Simplified 56.4%

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

       1. clear-num N/A

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

       2. 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 57.6%

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

   12. Applied egg-rr 57.6%

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

   if 1.7999999999999999e-14 < y < 4.0000000000000002e92 or 7.69999999999999994e239 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 16.2%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lowering-*.f64 29.5%

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

   8. Simplified 29.5%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. /-lowering-/.f64 60.9%

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

   10. Applied egg-rr 60.9%

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

   if 4.0000000000000002e92 < y < 7.69999999999999994e239

   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 81.1%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.0%

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

   10. Simplified 74.0%

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

Alternative 13: 70.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \frac{1}{\frac{1}{\frac{y + y \cdot \left(y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.05e+113)
   (*
    x
    (/
     1.0
     (/
      1.0
      (/
       (+
        y
        (*
         y
         (* y (* y (+ 0.16666666666666666 (* y (* y 0.008333333333333333)))))))
       x))))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.05e+113) {
		tmp = x * (1.0 / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x)));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.05d+113) then
        tmp = x * (1.0d0 / (1.0d0 / ((y + (y * (y * (y * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))))))) / x)))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.05e+113) {
		tmp = x * (1.0 / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x)));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.05e+113:
		tmp = x * (1.0 / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x)))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.05e+113)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 / Float64(Float64(y + Float64(y * Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333))))))) / x))));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.05e+113)
		tmp = x * (1.0 / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x)));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.05e+113], N[(x * N[(1.0 / N[(1.0 / N[(N[(y + N[(y * N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.04999999999999998e113

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.6%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 65.3%

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

   10. Simplified 65.3%

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

       1. div-inv N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. flip-+ N/A

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

       5. *-rgt-identity N/A

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

       6. fmm-def N/A

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

       7. frac-times N/A

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

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

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

   12. Applied egg-rr 27.6%

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

       1. clear-num N/A

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

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

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

       3. clear-num N/A

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

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

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

   14. Applied egg-rr 66.4%

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

   if 3.04999999999999998e113 < y

   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 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 81.6%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \frac{1}{\frac{1}{\frac{y + y \cdot \left(y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{y + y \cdot \left(y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+114)
   (/
    x
    (/
     1.0
     (/
      (+
       y
       (*
        y
        (* y (* y (+ 0.16666666666666666 (* y (* y 0.008333333333333333)))))))
      x)))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+114) {
		tmp = x / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.6d+114) then
        tmp = x / (1.0d0 / ((y + (y * (y * (y * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))))))) / x))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+114) {
		tmp = x / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+114:
		tmp = x / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+114)
		tmp = Float64(x / Float64(1.0 / Float64(Float64(y + Float64(y * Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333))))))) / x)));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+114)
		tmp = x / (1.0 / ((y + (y * (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+114], N[(x / N[(1.0 / N[(N[(y + N[(y * N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e114

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.6%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 65.3%

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

   10. Simplified 65.3%

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

       1. div-inv N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. flip-+ N/A

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

       5. *-rgt-identity N/A

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

       6. fmm-def N/A

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

       7. frac-times N/A

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

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

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

   12. Applied egg-rr 27.6%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

       4. clear-num N/A

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

   14. Applied egg-rr 66.4%

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

   if 2.6e114 < y

   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 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 81.6%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{y + y \cdot \left(y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.65e+48)
   (*
    x
    (/
     (*
      y
      (+
       1.0
       (* y (* y (+ 0.16666666666666666 (* y (* y 0.008333333333333333)))))))
     x))
   (if (<= x 5.5e+126)
     (*
      y
      (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))
     (* x (/ (* y (* 0.008333333333333333 (* y (* y (* y y))))) x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.65e+48) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x);
	} else if (x <= 5.5e+126) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (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 <= 1.65d+48) then
        tmp = x * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0))))))) / x)
    else if (x <= 5.5d+126) then
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    else
        tmp = x * ((y * (0.008333333333333333d0 * (y * (y * (y * y))))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.65e+48) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x);
	} else if (x <= 5.5e+126) {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	} else {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.65e+48:
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x)
	elif x <= 5.5e+126:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	else:
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.65e+48)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333))))))) / x));
	elseif (x <= 5.5e+126)
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	else
		tmp = Float64(x * Float64(Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * Float64(y * y))))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.65e+48)
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333))))))) / x);
	elseif (x <= 5.5e+126)
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	else
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.65e+48], N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+126], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(0.008333333333333333 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.65000000000000011e48

   1. Initial program 84.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.2%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 73.0%

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

   10. Simplified 73.0%

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

   if 1.65000000000000011e48 < x < 5.5000000000000004e126

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 40.3%

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

   5. Simplified 40.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 40.4%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 42.9%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 42.9%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]

   if 5.5000000000000004e126 < x

   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 51.0%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left({y}^{4}\right)\right)\right), x\right)\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left({y}^{3}\right)\right)\right)\right), x\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 55.0%

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

   13. Simplified 55.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{\frac{x + y \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)}{y}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4)
   (/ x (/ (+ x (* y (* y (* x -0.16666666666666666)))) y))
   (if (<= y 2.5e+113)
     (* x (/ (* y (* 0.008333333333333333 (* y (* y (* y y))))) x))
     (*
      y
      (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / ((x + (y * (y * (x * -0.16666666666666666)))) / y);
	} else if (y <= 2.5e+113) {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d0) then
        tmp = x / ((x + (y * (y * (x * (-0.16666666666666666d0))))) / y)
    else if (y <= 2.5d+113) then
        tmp = x * ((y * (0.008333333333333333d0 * (y * (y * (y * y))))) / x)
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4) {
		tmp = x / ((x + (y * (y * (x * -0.16666666666666666)))) / y);
	} else if (y <= 2.5e+113) {
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4:
		tmp = x / ((x + (y * (y * (x * -0.16666666666666666)))) / y)
	elif y <= 2.5e+113:
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x)
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4)
		tmp = Float64(x / Float64(Float64(x + Float64(y * Float64(y * Float64(x * -0.16666666666666666)))) / y));
	elseif (y <= 2.5e+113)
		tmp = Float64(x * Float64(Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * Float64(y * y))))) / x));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4)
		tmp = x / ((x + (y * (y * (x * -0.16666666666666666)))) / y);
	elseif (y <= 2.5e+113)
		tmp = x * ((y * (0.008333333333333333 * (y * (y * (y * y))))) / x);
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4], N[(x / N[(N[(x + N[(y * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+113], N[(x * N[(N[(y * N[(0.008333333333333333 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.39999999999999991

   1. Initial program 82.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 67.5%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.0%

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

   10. Simplified 66.0%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

       4. associate-/r* N/A

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

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

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

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

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

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

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. associate-*l* N/A

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

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

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

   12. Applied egg-rr 67.2%

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

   13. Taylor expanded in y around 0

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

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

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

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

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 49.8%

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

   15. Simplified 49.8%

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

   if 2.39999999999999991 < y < 2.5e113

   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 81.3%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 63.3%

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

   10. Simplified 63.3%

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

   11. Taylor expanded in y around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left({y}^{4}\right)\right)\right), x\right)\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left({y}^{3}\right)\right)\right)\right), x\right)\right) \]

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 63.3%

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

   13. Simplified 63.3%

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

   if 2.5e113 < y

   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 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. +-commutative N/A

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

   8. Simplified 81.6%

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

   9. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. distribute-rgt-in N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4:\\ \;\;\;\;\frac{x}{\frac{x + y \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)}{y}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x))))
   (if (<= x 1.65e+48)
     t_0
     (if (<= x 5.5e+126)
       (/ (* y (* -0.16666666666666666 (* x (* x x)))) x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.65e+48) {
		tmp = t_0;
	} else if (x <= 5.5e+126) {
		tmp = (y * (-0.16666666666666666 * (x * (x * x)))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
    if (x <= 1.65d+48) then
        tmp = t_0
    else if (x <= 5.5d+126) then
        tmp = (y * ((-0.16666666666666666d0) * (x * (x * x)))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	double tmp;
	if (x <= 1.65e+48) {
		tmp = t_0;
	} else if (x <= 5.5e+126) {
		tmp = (y * (-0.16666666666666666 * (x * (x * x)))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)
	tmp = 0
	if x <= 1.65e+48:
		tmp = t_0
	elif x <= 5.5e+126:
		tmp = (y * (-0.16666666666666666 * (x * (x * x)))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x))
	tmp = 0.0
	if (x <= 1.65e+48)
		tmp = t_0;
	elseif (x <= 5.5e+126)
		tmp = Float64(Float64(y * Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	tmp = 0.0;
	if (x <= 1.65e+48)
		tmp = t_0;
	elseif (x <= 5.5e+126)
		tmp = (y * (-0.16666666666666666 * (x * (x * x)))) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.65e+48], t$95$0, If[LessEqual[x, 5.5e+126], N[(N[(y * N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000011e48 or 5.5000000000000004e126 < x

   1. Initial program 86.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 73.1%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 68.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 68.4%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

   if 1.65000000000000011e48 < x < 5.5000000000000004e126

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \color{blue}{y}\right), x\right) \]
   4. Step-by-step derivation

   5. Simplified 55.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, y\right), x\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), y\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), y\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), y\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), y\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), y\right), x\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\mathsf{neg}\left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right), x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right), y\right), x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right), y\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right), y\right), x\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), y\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), y\right), x\right) \]

      13. *-lowering-*.f64 40.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), y\right), x\right) \]

   8. Simplified 40.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)} \cdot y}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{3}\right)}, y\right), x\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{3}\right)\right), y\right), x\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot x\right)\right)\right), y\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot {x}^{2}\right)\right), y\right), x\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), y\right), x\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), y\right), x\right) \]

       6. *-lowering-*.f64 40.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), x\right) \]

   11. Simplified 40.3%

       \[\leadsto \frac{\color{blue}{\left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \cdot y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+113)
   (/ x (/ (/ x y) (+ 1.0 (* y (* y 0.16666666666666666)))))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+113) {
		tmp = x / ((x / y) / (1.0 + (y * (y * 0.16666666666666666))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d+113) then
        tmp = x / ((x / y) / (1.0d0 + (y * (y * 0.16666666666666666d0))))
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e+113) {
		tmp = x / ((x / y) / (1.0 + (y * (y * 0.16666666666666666))));
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+113:
		tmp = x / ((x / y) / (1.0 + (y * (y * 0.16666666666666666))))
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+113)
		tmp = Float64(x / Float64(Float64(x / y) / Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)))));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+113)
		tmp = x / ((x / y) / (1.0 + (y * (y * 0.16666666666666666))));
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+113], N[(x / N[(N[(x / y), $MachinePrecision] / N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e113

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}, x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right), x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right), x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

      14. *-lowering-*.f64 66.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}{x} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{x}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{x}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}\right)}\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\frac{x}{y}}{\color{blue}{1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\color{blue}{1} + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}\right)\right)\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}}} \]

   13. Taylor expanded in y around 0

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]
   14. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 63.8%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right) \]

   15. Simplified 63.8%

       \[\leadsto \frac{x}{\frac{\frac{x}{y}}{\color{blue}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}} \]

   if 2e113 < y

   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 81.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 81.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 81.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{6} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{\frac{\frac{x}{y}}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+114)
   (* x (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) x))
   (*
    y
    (* (* y y) (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+114) {
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+114) then
        tmp = x * ((y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / x)
    else
        tmp = y * ((y * y) * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+114) {
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	} else {
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+114:
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x)
	else:
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+114)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / x));
	else
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+114)
		tmp = x * ((y * (1.0 + (0.16666666666666666 * (y * y)))) / x);
	else
		tmp = y * ((y * y) * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+114], N[(x * N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e114

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 68.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 63.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 63.1%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}{x} \]

   if 5.0000000000000001e114 < y

   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 81.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right), \mathsf{sinh.f64}\left(y\right)\right), x\right) \]

   5. Simplified 81.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\right)} \cdot \sinh y}{x} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + \frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto y \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{y \cdot \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto y \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + y \cdot \left(\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{y}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot {y}^{2}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot {y}^{2}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

   8. Simplified 81.6%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot \left({y}^{2} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right)\right) \]

       7. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{6} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{6} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{-1}{36} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\frac{-1}{36}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   11. Simplified 81.6%

       \[\leadsto y \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 56.7% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e+92) (/ x (/ x y)) (* y (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+92) {
		tmp = x / (x / y);
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d+92) then
        tmp = x / (x / y)
    else
        tmp = y * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e+92) {
		tmp = x / (x / y);
	} else {
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+92:
		tmp = x / (x / y)
	else:
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+92)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e+92)
		tmp = x / (x / y);
	else
		tmp = y * (1.0 + (0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+92], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000002e92

   1. Initial program 83.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   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 67.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 54.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 54.5%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

       2. 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 55.7%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   12. Applied egg-rr 55.7%

       \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]

   if 4.0000000000000002e92 < 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 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 78.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (/ x y)))

C

double code(double x, double y) {
	return x / (x / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (x / y)
end function

Java

public static double code(double x, double y) {
	return x / (x / y);
}

Python

def code(x, y):
	return x / (x / y)

Julia

function code(x, y)
	return Float64(x / Float64(x / y))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (x / y);
end

Wolfram

code[x_, y_] := N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
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 69.9%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

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 50.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 50.4%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{x}{y}}} \]

    2. 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 51.3%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

12. Applied egg-rr 51.3%

    \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 22: 50.5% 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 87.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
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 69.9%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

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 50.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 50.4%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 23: 28.4% 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 87.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
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 69.9%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
9. Step-by-step derivation

10. Simplified 29.7%

    \[\leadsto \color{blue}{y} \]
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 2024138 
(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))