Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.8%

Time: 15.6s

Alternatives: 21

Speedup: 1.6×

• 49.5% 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: 89.1% 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: 99.8% 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 87.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-26}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7e-26)
   (sinh y)
   (/
    (*
     (sin x)
     (*
      y
      (+
       1.0
       (*
        (* y y)
        (+
         0.16666666666666666
         (*
          y
          (*
           y
           (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7e-26) {
		tmp = sinh(y);
	} else {
		tmp = (sin(x) * (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) / 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 <= 7d-26) then
        tmp = sinh(y)
    else
        tmp = (sin(x) * (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7e-26) {
		tmp = Math.sinh(y);
	} else {
		tmp = (Math.sin(x) * (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7e-26:
		tmp = math.sinh(y)
	else:
		tmp = (math.sin(x) * (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7e-26)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(sin(x) * Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7e-26)
		tmp = sinh(y);
	else
		tmp = (sin(x) * (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7e-26], N[Sinh[y], $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999997e-26

   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 99.9%

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

   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 85.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 80.0%

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

   9. Applied egg-rr 80.0%

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

   if 6.9999999999999997e-26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 95.6%

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

   5. Simplified 95.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-26}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e-58)
   (sinh y)
   (*
    (*
     y
     (+
      1.0
      (*
       (* y y)
       (+
        0.16666666666666666
        (*
         (* y y)
         (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
    (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e-58) {
		tmp = sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) * (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 (x <= 2d-58) then
        tmp = sinh(y)
    else
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))) * (sin(x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e-58) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e-58:
		tmp = math.sinh(y)
	else:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e-58)
		tmp = sinh(y);
	else
		tmp = 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))))))) * Float64(sin(x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2e-58)
		tmp = sinh(y);
	else
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))) * (sin(x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e-58], N[Sinh[y], $MachinePrecision], 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] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-58

   1. Initial program 83.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 85.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

   if 2.0000000000000001e-58 < x

   1. Initial program 98.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-lft-in N/A

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

   7. Simplified 94.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 94.7%

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

   10. Simplified 94.7%

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

   11. Taylor expanded in x around inf

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

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

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

   13. Simplified 94.7%

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

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

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

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

   15. Applied egg-rr 95.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \frac{\sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (sin x)
          (/
           (*
            y
            (+
             1.0
             (*
              y
              (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))
           x))))
   (if (<= y 1500000.0) t_0 (if (<= y 1.2e+62) (sinh y) t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	double tmp;
	if (y <= 1500000.0) {
		tmp = t_0;
	} else if (y <= 1.2e+62) {
		tmp = sinh(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 = sin(x) * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)))))) / x)
    if (y <= 1500000.0d0) then
        tmp = t_0
    else if (y <= 1.2d+62) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	double tmp;
	if (y <= 1500000.0) {
		tmp = t_0;
	} else if (y <= 1.2e+62) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x)
	tmp = 0
	if y <= 1500000.0:
		tmp = t_0
	elif y <= 1.2e+62:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) / x))
	tmp = 0.0
	if (y <= 1500000.0)
		tmp = t_0;
	elseif (y <= 1.2e+62)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * ((y * (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) / x);
	tmp = 0.0;
	if (y <= 1500000.0)
		tmp = t_0;
	elseif (y <= 1.2e+62)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1500000.0], t$95$0, If[LessEqual[y, 1.2e+62], N[Sinh[y], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e6 or 1.2e62 < y

   1. Initial program 87.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.7%

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

   7. Simplified 96.7%

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

   if 1.5e6 < y < 1.2e62

   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 83.3%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 83.3%

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

   9. Applied egg-rr 83.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;\sin x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot t\_0\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+103}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot t\_0\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* y y) 0.16666666666666666))))
   (if (<= y 1500000.0)
     (* y (* (/ (sin x) x) t_0))
     (if (<= y 1.35e+103) (sinh y) (/ (* y (* (sin x) t_0)) x)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 1500000.0) {
		tmp = y * ((sin(x) / x) * t_0);
	} else if (y <= 1.35e+103) {
		tmp = sinh(y);
	} else {
		tmp = (y * (sin(x) * t_0)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    if (y <= 1500000.0d0) then
        tmp = y * ((sin(x) / x) * t_0)
    else if (y <= 1.35d+103) then
        tmp = sinh(y)
    else
        tmp = (y * (sin(x) * t_0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 1500000.0) {
		tmp = y * ((Math.sin(x) / x) * t_0);
	} else if (y <= 1.35e+103) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (Math.sin(x) * t_0)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * y) * 0.16666666666666666)
	tmp = 0
	if y <= 1500000.0:
		tmp = y * ((math.sin(x) / x) * t_0)
	elif y <= 1.35e+103:
		tmp = math.sinh(y)
	else:
		tmp = (y * (math.sin(x) * t_0)) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))
	tmp = 0.0
	if (y <= 1500000.0)
		tmp = Float64(y * Float64(Float64(sin(x) / x) * t_0));
	elseif (y <= 1.35e+103)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * Float64(sin(x) * t_0)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	tmp = 0.0;
	if (y <= 1500000.0)
		tmp = y * ((sin(x) / x) * t_0);
	elseif (y <= 1.35e+103)
		tmp = sinh(y);
	else
		tmp = (y * (sin(x) * t_0)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1500000.0], N[(y * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+103], N[Sinh[y], $MachinePrecision], N[(N[(y * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5e6

   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 88.3%

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

   if 1.5e6 < y < 1.34999999999999996e103

   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 85.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 85.7%

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

   9. Applied egg-rr 85.7%

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

   if 1.34999999999999996e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+103}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\sin x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\frac{\sin x}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (/ (sin x) x) (+ 1.0 (* (* y y) 0.16666666666666666))))))
   (if (<= y 1500000.0) t_0 (if (<= y 2.5e+114) (sinh y) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((sin(x) / x) * (1.0 + ((y * y) * 0.16666666666666666)));
	double tmp;
	if (y <= 1500000.0) {
		tmp = t_0;
	} else if (y <= 2.5e+114) {
		tmp = sinh(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 = y * ((sin(x) / x) * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
    if (y <= 1500000.0d0) then
        tmp = t_0
    else if (y <= 2.5d+114) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((Math.sin(x) / x) * (1.0 + ((y * y) * 0.16666666666666666)));
	double tmp;
	if (y <= 1500000.0) {
		tmp = t_0;
	} else if (y <= 2.5e+114) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((math.sin(x) / x) * (1.0 + ((y * y) * 0.16666666666666666)))
	tmp = 0
	if y <= 1500000.0:
		tmp = t_0
	elif y <= 2.5e+114:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(sin(x) / x) * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))))
	tmp = 0.0
	if (y <= 1500000.0)
		tmp = t_0;
	elseif (y <= 2.5e+114)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((sin(x) / x) * (1.0 + ((y * y) * 0.16666666666666666)));
	tmp = 0.0;
	if (y <= 1500000.0)
		tmp = t_0;
	elseif (y <= 2.5e+114)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1500000.0], t$95$0, If[LessEqual[y, 2.5e+114], N[Sinh[y], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e6 or 2.5e114 < y

   1. Initial program 86.7%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      2. fma-define N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. fma-define N/A

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

      8. distribute-lft-in N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

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

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

   7. Simplified 89.0%

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

   if 1.5e6 < y < 2.5e114

   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 87.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 87.5%

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

   9. Applied egg-rr 87.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{\sin x}{x} \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1500000.0)
   (* y (/ (sin x) x))
   (if (<= y 2e+114)
     (sinh y)
     (*
      (* y (* y y))
      (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1500000.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 2e+114) {
		tmp = sinh(y);
	} 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 <= 1500000.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 2d+114) then
        tmp = sinh(y)
    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 <= 1500000.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 2e+114) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1500000.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 2e+114:
		tmp = math.sinh(y)
	else:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1500000.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 2e+114)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * 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 <= 1500000.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 2e+114)
		tmp = sinh(y);
	else
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1500000.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+114], N[Sinh[y], $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5e6

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 51.1%

      \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{x} \]
   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. *-lowering-*.f64 N/A

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

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

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

      5. sin-lowering-sin.f64 67.1%

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

   7. Applied egg-rr 67.1%

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

   if 1.5e6 < y < 2e114

   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 87.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 87.5%

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

   9. Applied egg-rr 87.5%

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

   if 2e114 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

   8. Simplified 84.6%

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

   9. Taylor expanded in y around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.6%

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

   11. Simplified 84.6%

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

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 84.6%

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

   14. Simplified 84.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1500000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-51)
   (/ x (/ x y))
   (if (<= y 2e+114)
     (sinh y)
     (*
      (* y (* y y))
      (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-51) {
		tmp = x / (x / y);
	} else if (y <= 2e+114) {
		tmp = sinh(y);
	} 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 <= 3.4d-51) then
        tmp = x / (x / y)
    else if (y <= 2d+114) then
        tmp = sinh(y)
    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 <= 3.4e-51) {
		tmp = x / (x / y);
	} else if (y <= 2e+114) {
		tmp = Math.sinh(y);
	} else {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.4e-51:
		tmp = x / (x / y)
	elif y <= 2e+114:
		tmp = math.sinh(y)
	else:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-51)
		tmp = Float64(x / Float64(x / y));
	elseif (y <= 2e+114)
		tmp = sinh(y);
	else
		tmp = Float64(Float64(y * 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 <= 3.4e-51)
		tmp = x / (x / y);
	elseif (y <= 2e+114)
		tmp = sinh(y);
	else
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.4e-51], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+114], N[Sinh[y], $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.40000000000000003e-51

   1. Initial program 83.2%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 78.4%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 62.9%

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

   10. Simplified 62.9%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

       4. /-lowering-/.f64 62.1%

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

   12. Applied egg-rr 62.1%

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

   if 3.40000000000000003e-51 < y < 2e114

   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 70.6%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

         \[\leadsto \sinh y \cdot {x}^{0} \]

      7. metadata-eval N/A

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

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

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

      9. sinh-lowering-sinh.f64 70.6%

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

   9. Applied egg-rr 70.6%

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

   if 2e114 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

   8. Simplified 84.6%

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

   9. Taylor expanded in y around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.6%

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

   11. Simplified 84.6%

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

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 84.6%

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

   14. Simplified 84.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e+114)
   (* x (/ (sinh y) x))
   (*
    (* y (* y y))
    (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+114) {
		tmp = x * (sinh(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 <= 2.25d+114) then
        tmp = x * (sinh(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 <= 2.25e+114) {
		tmp = x * (Math.sinh(y) / x);
	} else {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.25e+114:
		tmp = x * (math.sinh(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 <= 2.25e+114)
		tmp = Float64(x * Float64(sinh(y) / x));
	else
		tmp = Float64(Float64(y * 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 <= 2.25e+114)
		tmp = x * (sinh(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, 2.25e+114], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e114

   1. Initial program 85.7%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 77.2%

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

   if 2.25e114 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

   8. Simplified 84.6%

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

   9. Taylor expanded in y around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.6%

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

   11. Simplified 84.6%

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

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 84.6%

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

   14. Simplified 84.6%

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

Alternative 10: 60.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.8e+42)
   (* x (/ 1.0 (/ x y)))
   (if (<= y 2e+114)
     (*
      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 <= 5.8e+42) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 2e+114) {
		tmp = 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 <= 5.8d+42) then
        tmp = x * (1.0d0 / (x / y))
    else if (y <= 2d+114) then
        tmp = 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 <= 5.8e+42) {
		tmp = x * (1.0 / (x / y));
	} else if (y <= 2e+114) {
		tmp = 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 <= 5.8e+42:
		tmp = x * (1.0 / (x / y))
	elif y <= 2e+114:
		tmp = 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 <= 5.8e+42)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	elseif (y <= 2e+114)
		tmp = Float64(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(Float64(y * 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 <= 5.8e+42)
		tmp = x * (1.0 / (x / y));
	elseif (y <= 2e+114)
		tmp = 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, 5.8e+42], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+114], N[(y * 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], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.79999999999999961e42

   1. Initial program 84.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.1%

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

   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 59.9%

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

   10. Simplified 59.9%

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

       1. clear-num N/A

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

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

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

       3. /-lowering-/.f64 60.6%

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

   12. Applied egg-rr 60.6%

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

   if 5.79999999999999961e42 < y < 2e114

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-lft-in N/A

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 88.9%

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

   10. Simplified 88.9%

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

   if 2e114 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

   8. Simplified 84.6%

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

   9. Taylor expanded in y around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 84.6%

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

   11. Simplified 84.6%

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

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

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

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

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

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

       8. unpow2 N/A

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

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

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

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

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. metadata-eval N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 84.6%

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

   14. Simplified 84.6%

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

Alternative 11: 71.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+114)
   (*
    x
    (/
     (*
      y
      (+
       1.0
       (*
        y
        (*
         y
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (+ 0.008333333333333333 (* y (* y 0.0001984126984126984))))))))))
     x))
   (*
    (* y (* y y))
    (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+114) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))))))))) / 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 <= 2d+114) then
        tmp = x * ((y * (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0)))))))))) / 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 <= 2e+114) {
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))))))))) / x);
	} else {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+114:
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))))))))) / x)
	else:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+114)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984)))))))))) / x));
	else
		tmp = Float64(Float64(y * 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+114)
		tmp = x * ((y * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))))))))) / x);
	else
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+114], N[(x * N[(N[(y * 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] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e114

   1. Initial program 85.7%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 77.2%

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

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 74.1%

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

   10. Simplified 74.1%

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

   if 2e114 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. *-commutative N/A

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

   8. Simplified 84.6%

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

   9. Taylor expanded in y around inf

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

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 84.6%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       18. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   14. Simplified 84.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+114}:\\ \;\;\;\;y \cdot \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+59)
   (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
   (if (<= y 2.15e+114)
     (*
      y
      (*
       (* (* y y) (* y y))
       (+ 0.008333333333333333 (/ 0.16666666666666666 (* y y)))))
     (*
      (* y (* y y))
      (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (y <= 2.15e+114) {
		tmp = y * (((y * y) * (y * y)) * (0.008333333333333333 + (0.16666666666666666 / (y * y))));
	} 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 <= 7.2d+59) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else if (y <= 2.15d+114) then
        tmp = y * (((y * y) * (y * y)) * (0.008333333333333333d0 + (0.16666666666666666d0 / (y * y))))
    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 <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (y <= 2.15e+114) {
		tmp = y * (((y * y) * (y * y)) * (0.008333333333333333 + (0.16666666666666666 / (y * y))));
	} else {
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+59:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	elif y <= 2.15e+114:
		tmp = y * (((y * y) * (y * y)) * (0.008333333333333333 + (0.16666666666666666 / (y * y))))
	else:
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+59)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	elseif (y <= 2.15e+114)
		tmp = Float64(y * Float64(Float64(Float64(y * y) * Float64(y * y)) * Float64(0.008333333333333333 + Float64(0.16666666666666666 / Float64(y * y)))));
	else
		tmp = Float64(Float64(y * 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 <= 7.2e+59)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	elseif (y <= 2.15e+114)
		tmp = y * (((y * y) * (y * y)) * (0.008333333333333333 + (0.16666666666666666 / (y * y))));
	else
		tmp = (y * (y * y)) * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+59], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+114], N[(y * N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.008333333333333333 + N[(0.16666666666666666 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.1999999999999997e59

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   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 76.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}}{x} \]

   if 7.1999999999999997e59 < y < 2.15e114

   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 91.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 91.7%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 91.7%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 91.7%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 91.7%

       \[\leadsto y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)} \]

   if 2.15e114 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 84.6%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 84.6%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       18. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   14. Simplified 84.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.35e+114)
   (*
    x
    (/
     (*
      y
      (+
       1.0
       (* 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 <= 2.35e+114) {
		tmp = x * ((y * (1.0 + (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 <= 2.35d+114) then
        tmp = x * ((y * (1.0d0 + (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 <= 2.35e+114) {
		tmp = x * ((y * (1.0 + (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 <= 2.35e+114:
		tmp = x * ((y * (1.0 + (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 <= 2.35e+114)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) / x));
	else
		tmp = Float64(Float64(y * 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 <= 2.35e+114)
		tmp = x * ((y * (1.0 + (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, 2.35e+114], N[(x * N[(N[(y * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.35e114

   1. Initial program 85.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \sin x \cdot \color{blue}{\frac{\sinh y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{\sinh y}{x}\right)}\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\color{blue}{\sinh y}}{x}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\sinh y, \color{blue}{x}\right)\right) \]

      5. sinh-lowering-sinh.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 77.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

      11. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), x\right)\right) \]

   10. Simplified 71.6%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}{x} \]

   if 2.35e114 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 84.6%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 84.6%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       18. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   14. Simplified 84.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+114}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y y))))
   (if (<= y 7.2e+59)
     (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
     (if (<= y 2.25e+114)
       (* y (* y (* 0.008333333333333333 t_0)))
       (* t_0 (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (y <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (y <= 2.25e+114) {
		tmp = y * (y * (0.008333333333333333 * t_0));
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * y)
    if (y <= 7.2d+59) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else if (y <= 2.25d+114) then
        tmp = y * (y * (0.008333333333333333d0 * t_0))
    else
        tmp = t_0 * (0.16666666666666666d0 + ((x * x) * (-0.027777777777777776d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * y);
	double tmp;
	if (y <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else if (y <= 2.25e+114) {
		tmp = y * (y * (0.008333333333333333 * t_0));
	} else {
		tmp = t_0 * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * y)
	tmp = 0
	if y <= 7.2e+59:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	elif y <= 2.25e+114:
		tmp = y * (y * (0.008333333333333333 * t_0))
	else:
		tmp = t_0 * (0.16666666666666666 + ((x * x) * -0.027777777777777776))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * y))
	tmp = 0.0
	if (y <= 7.2e+59)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	elseif (y <= 2.25e+114)
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * t_0)));
	else
		tmp = Float64(t_0 * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.027777777777777776)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * y);
	tmp = 0.0;
	if (y <= 7.2e+59)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	elseif (y <= 2.25e+114)
		tmp = y * (y * (0.008333333333333333 * t_0));
	else
		tmp = t_0 * (0.16666666666666666 + ((x * x) * -0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e+59], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+114], N[(y * N[(y * N[(0.008333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.1999999999999997e59

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   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 76.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}}{x} \]

   if 7.1999999999999997e59 < y < 2.25e114

   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 91.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 91.7%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 91.7%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 91.7%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 91.7%

       \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]

   if 2.25e114 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + {y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x + \frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right), x\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left({y}^{2} \cdot \left(\frac{1}{6} \cdot \sin x\right) + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left({y}^{2} \cdot \frac{1}{6}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right)\right)\right)\right), x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + {y}^{2} \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x\right)\right)\right)\right), x\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \sin x + \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot \sin x\right)\right)\right), x\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left(\frac{1}{6} \cdot {y}^{2} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

      11. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right), x\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({x}^{2} \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) + y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) + \color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \]

      2. distribute-lft1-in N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), \left(\color{blue}{y} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({x}^{2}\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 84.6%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 84.6%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)}\right) \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{3} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{3}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left({y}^{3} \cdot \frac{1}{6}\right) \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {x}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto {y}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{3}\right), \color{blue}{\left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}\right) \]

       5. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot y\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot {y}^{2}\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{6}} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)\right) \]

       10. distribute-rgt-in N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(1 \cdot \frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}}\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{6} + \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)} \cdot \frac{1}{6}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot \frac{1}{6}\right)}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left({x}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{1}{6}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \frac{-1}{36}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right) \]

       18. *-lowering-*.f64 84.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right) \]

   14. Simplified 84.6%

       \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+59)
   (* x (/ (* y (+ 1.0 (* (* y y) 0.16666666666666666))) x))
   (* y (* y (* 0.008333333333333333 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * (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 <= 7.2d+59) then
        tmp = x * ((y * (1.0d0 + ((y * y) * 0.16666666666666666d0))) / x)
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+59) {
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+59:
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x)
	else:
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+59)
		tmp = Float64(x * Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))) / x));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e+59)
		tmp = x * ((y * (1.0 + ((y * y) * 0.16666666666666666))) / x);
	else
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+59], N[(x * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.1999999999999997e59

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   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 76.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), x\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

      6. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right), x\right)\right) \]

   10. Simplified 66.7%

       \[\leadsto x \cdot \frac{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}}{x} \]

   if 7.1999999999999997e59 < 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 78.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 78.4%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 78.4%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 78.4%

       \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 59.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e+59)
   (* x (/ 1.0 (/ x y)))
   (* y (* y (* 0.008333333333333333 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+59) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d+59) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * (y * (0.008333333333333333d0 * (y * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e+59) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e+59:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e+59)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(y * Float64(0.008333333333333333 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e+59)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * (y * (0.008333333333333333 * (y * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e+59], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(0.008333333333333333 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.04999999999999992e59

   1. Initial program 84.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

   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 76.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 58.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 58.7%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 59.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 59.4%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]

   if 1.04999999999999992e59 < 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 78.4%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 78.4%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)}\right) \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

       11. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 78.4%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 78.4%

       \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(0.008333333333333333 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e+102)
   (* x (/ 1.0 (/ x y)))
   (* y (+ 1.0 (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e+102) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d+102) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e+102) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e+102:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * (1.0 + (y * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e+102)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e+102)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * (1.0 + (y * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e+102], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999982e102

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.9%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 58.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 58.6%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 59.3%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]

   if 7.99999999999999982e102 < 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 76.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 76.2%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   13. Simplified 76.2%

       \[\leadsto y \cdot \left(1 + y \cdot \color{blue}{\left(y \cdot 0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 57.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e+103)
   (* x (/ 1.0 (/ x y)))
   (* y (* y (* y 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+103) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.02d+103) then
        tmp = x * (1.0d0 / (x / y))
    else
        tmp = y * (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+103) {
		tmp = x * (1.0 / (x / y));
	} else {
		tmp = y * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.02e+103:
		tmp = x * (1.0 / (x / y))
	else:
		tmp = y * (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.02e+103)
		tmp = Float64(x * Float64(1.0 / Float64(x / y)));
	else
		tmp = Float64(y * Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e+103)
		tmp = x * (1.0 / (x / y));
	else
		tmp = y * (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.02e+103], N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.01999999999999991e103

   1. Initial program 85.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.9%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   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 58.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

   10. Simplified 58.6%

       \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

       3. /-lowering-/.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   12. Applied egg-rr 59.3%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]

   if 1.01999999999999991e103 < 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 76.2%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 76.2%

       \[\leadsto \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{4} \cdot \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{120} + \frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{{y}^{2}}\right)}\right)\right)\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6} \cdot 1}{\color{blue}{{y}^{2}}}\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{\frac{1}{6}}{{\color{blue}{y}}^{2}}\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 76.2%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{/.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   13. Simplified 76.2%

       \[\leadsto y \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(0.008333333333333333 + \frac{0.16666666666666666}{y \cdot y}\right)\right)} \]

   14. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{3}} \]
   15. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left({y}^{2} \cdot y\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y} \]

       4. *-commutative N/A

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{y}\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

       12. *-lowering-*.f64 76.2%

           \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   16. Simplified 76.2%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 51.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ 1.0 (/ x y))))

C

double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 / (x / y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 / (x / y));
}

Python

def code(x, y):
	return x * (1.0 / (x / y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 / Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 / (x / y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

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 76.8%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 55.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 55.6%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{x}{y}}}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

    3. /-lowering-/.f64 56.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

12. Applied egg-rr 56.1%

    \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{x}{y}}} \]
13. Add Preprocessing

Alternative 20: 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 87.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

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 76.8%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y}{x}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 55.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right) \]

10. Simplified 55.6%

    \[\leadsto x \cdot \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 21: 27.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 87.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), x\right)\right) \]

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 76.8%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y} \]
9. Step-by-step derivation

10. Simplified 28.8%

    \[\leadsto \color{blue}{y} \]
11. Add Preprocessing

Developer Target 1: 99.8% 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 2024152 
(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))