Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 18.4s

Alternatives: 21

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 343681.3893244343
  2. y = 1.281665463547106e+289

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

   6. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 3: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot \frac{2}{y}\right) \cdot \frac{\sinh y}{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-19)
   (sin x)
   (if (<= y 3.8e+77)
     (* (* x (/ 2.0 y)) (/ (sinh y) 2.0))
     (* (sin x) (* y (* y (* (* y y) 0.008333333333333333)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = sin(x);
	} else if (y <= 3.8e+77) {
		tmp = (x * (2.0 / y)) * (sinh(y) / 2.0);
	} else {
		tmp = sin(x) * (y * (y * ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.9d-19) then
        tmp = sin(x)
    else if (y <= 3.8d+77) then
        tmp = (x * (2.0d0 / y)) * (sinh(y) / 2.0d0)
    else
        tmp = sin(x) * (y * (y * ((y * y) * 0.008333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = Math.sin(x);
	} else if (y <= 3.8e+77) {
		tmp = (x * (2.0 / y)) * (Math.sinh(y) / 2.0);
	} else {
		tmp = Math.sin(x) * (y * (y * ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.9e-19:
		tmp = math.sin(x)
	elif y <= 3.8e+77:
		tmp = (x * (2.0 / y)) * (math.sinh(y) / 2.0)
	else:
		tmp = math.sin(x) * (y * (y * ((y * y) * 0.008333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 3.8e+77)
		tmp = Float64(Float64(x * Float64(2.0 / y)) * Float64(sinh(y) / 2.0));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 3.8e+77)
		tmp = (x * (2.0 / y)) * (sinh(y) / 2.0);
	else
		tmp = sin(x) * (y * (y * ((y * y) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.9e-19], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[(x * N[(2.0 / y), $MachinePrecision]), $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.90000000000000038e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.1%

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

   5. Simplified 70.1%

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

   if 5.90000000000000038e-19 < y < 3.8000000000000001e77

   1. Initial program 99.8%

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

      1. sinh-def N/A

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

      2. associate-/l/ N/A

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

      3. sinh-undef N/A

         \[\leadsto \sin x \cdot \frac{2 \cdot \sinh y}{\color{blue}{y} \cdot 2} \]

      4. times-frac N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 92.3%

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

   if 3.8000000000000001e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 4: 71.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-19)
   (sin x)
   (if (<= y 4.5e+146)
     (/ x (/ y (sinh y)))
     (* (sin x) (+ 1.0 (* y (* y 0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = sin(x);
	} else if (y <= 4.5e+146) {
		tmp = x / (y / sinh(y));
	} else {
		tmp = sin(x) * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.9d-19) then
        tmp = sin(x)
    else if (y <= 4.5d+146) then
        tmp = x / (y / sinh(y))
    else
        tmp = sin(x) * (1.0d0 + (y * (y * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = Math.sin(x);
	} else if (y <= 4.5e+146) {
		tmp = x / (y / Math.sinh(y));
	} else {
		tmp = Math.sin(x) * (1.0 + (y * (y * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.9e-19:
		tmp = math.sin(x)
	elif y <= 4.5e+146:
		tmp = x / (y / math.sinh(y))
	else:
		tmp = math.sin(x) * (1.0 + (y * (y * 0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 4.5e+146)
		tmp = Float64(x / Float64(y / sinh(y)));
	else
		tmp = Float64(sin(x) * 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 <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 4.5e+146)
		tmp = x / (y / sinh(y));
	else
		tmp = sin(x) * (1.0 + (y * (y * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.9e-19], N[Sin[x], $MachinePrecision], If[LessEqual[y, 4.5e+146], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.90000000000000038e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.1%

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

   5. Simplified 70.1%

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

   if 5.90000000000000038e-19 < y < 4.50000000000000026e146

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 87.0%

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

   if 4.50000000000000026e146 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 97.8%

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

   5. Simplified 97.8%

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

Alternative 5: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-19)
   (sin x)
   (if (<= y 4.5e+146)
     (/ x (/ y (sinh y)))
     (* (sin x) (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = sin(x);
	} else if (y <= 4.5e+146) {
		tmp = x / (y / sinh(y));
	} else {
		tmp = sin(x) * ((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 <= 5.9d-19) then
        tmp = sin(x)
    else if (y <= 4.5d+146) then
        tmp = x / (y / sinh(y))
    else
        tmp = sin(x) * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = Math.sin(x);
	} else if (y <= 4.5e+146) {
		tmp = x / (y / Math.sinh(y));
	} else {
		tmp = Math.sin(x) * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.9e-19:
		tmp = math.sin(x)
	elif y <= 4.5e+146:
		tmp = x / (y / math.sinh(y))
	else:
		tmp = math.sin(x) * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 4.5e+146)
		tmp = Float64(x / Float64(y / sinh(y)));
	else
		tmp = Float64(sin(x) * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 4.5e+146)
		tmp = x / (y / sinh(y));
	else
		tmp = sin(x) * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.9e-19], N[Sin[x], $MachinePrecision], If[LessEqual[y, 4.5e+146], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.90000000000000038e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.1%

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

   5. Simplified 70.1%

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

   if 5.90000000000000038e-19 < y < 4.50000000000000026e146

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 87.0%

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

   if 4.50000000000000026e146 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 97.8%

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

   8. Simplified 97.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+254}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-19)
   (sin x)
   (if (<= y 2e+254)
     (/ x (/ y (sinh y)))
     (*
      (* y (* x (+ 0.008333333333333333 (* (* x x) -0.001388888888888889))))
      (* y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = sin(x);
	} else if (y <= 2e+254) {
		tmp = x / (y / sinh(y));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (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 <= 5.9d-19) then
        tmp = sin(x)
    else if (y <= 2d+254) then
        tmp = x / (y / sinh(y))
    else
        tmp = (y * (x * (0.008333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))) * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = Math.sin(x);
	} else if (y <= 2e+254) {
		tmp = x / (y / Math.sinh(y));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.9e-19:
		tmp = math.sin(x)
	elif y <= 2e+254:
		tmp = x / (y / math.sinh(y))
	else:
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 2e+254)
		tmp = Float64(x / Float64(y / sinh(y)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.001388888888888889)))) * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 2e+254)
		tmp = x / (y / sinh(y));
	else
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.9e-19], N[Sin[x], $MachinePrecision], If[LessEqual[y, 2e+254], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.90000000000000038e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.1%

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

   5. Simplified 70.1%

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

   if 5.90000000000000038e-19 < y < 1.9999999999999999e254

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 83.0%

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

   if 1.9999999999999999e254 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot {y}^{4}\right) + x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \color{blue}{x} \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{{y}^{4}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(\frac{1}{120} \cdot x\right) \cdot {\color{blue}{y}}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{120}\right)}\right)\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{720} \cdot {x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. cube-unmult N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot {y}^{\color{blue}{3}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right) \cdot \color{blue}{{y}^{3}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right), \color{blue}{\left({y}^{3}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({\color{blue}{y}}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+254}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.95 \cdot 10^{+254}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-19)
   (sin x)
   (if (<= y 3.95e+254)
     (* x (/ (sinh y) y))
     (*
      (* y (* x (+ 0.008333333333333333 (* (* x x) -0.001388888888888889))))
      (* y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = sin(x);
	} else if (y <= 3.95e+254) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (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 <= 5.9d-19) then
        tmp = sin(x)
    else if (y <= 3.95d+254) then
        tmp = x * (sinh(y) / y)
    else
        tmp = (y * (x * (0.008333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))) * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-19) {
		tmp = Math.sin(x);
	} else if (y <= 3.95e+254) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.9e-19:
		tmp = math.sin(x)
	elif y <= 3.95e+254:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 3.95e+254)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.001388888888888889)))) * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.9e-19)
		tmp = sin(x);
	elseif (y <= 3.95e+254)
		tmp = x * (sinh(y) / y);
	else
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.9e-19], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.95e+254], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.90000000000000038e-19

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.1%

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

   5. Simplified 70.1%

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

   if 5.90000000000000038e-19 < y < 3.9499999999999999e254

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 83.0%

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

   if 3.9499999999999999e254 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot {y}^{4}\right) + x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \color{blue}{x} \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{{y}^{4}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(\frac{1}{120} \cdot x\right) \cdot {\color{blue}{y}}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{120}\right)}\right)\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{720} \cdot {x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. cube-unmult N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot {y}^{\color{blue}{3}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right) \cdot \color{blue}{{y}^{3}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right), \color{blue}{\left({y}^{3}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({\color{blue}{y}}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-19}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.95 \cdot 10^{+254}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00095:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00095)
   (sin x)
   (if (<= y 2e+252)
     (/
      (+
       1.0
       (*
        (* y y)
        (+
         0.16666666666666666
         (*
          (* y y)
          (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))
      (/ 1.0 x))
     (*
      (* y (* x (+ 0.008333333333333333 (* (* x x) -0.001388888888888889))))
      (* y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00095) {
		tmp = sin(x);
	} else if (y <= 2e+252) {
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / (1.0 / x);
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (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 <= 0.00095d0) then
        tmp = sin(x)
    else if (y <= 2d+252) then
        tmp = (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))) / (1.0d0 / x)
    else
        tmp = (y * (x * (0.008333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))) * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00095) {
		tmp = Math.sin(x);
	} else if (y <= 2e+252) {
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / (1.0 / x);
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00095:
		tmp = math.sin(x)
	elif y <= 2e+252:
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / (1.0 / x)
	else:
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00095)
		tmp = sin(x);
	elseif (y <= 2e+252)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))))) / Float64(1.0 / x));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.001388888888888889)))) * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00095)
		tmp = sin(x);
	elseif (y <= 2e+252)
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / (1.0 / x);
	else
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00095], N[Sin[x], $MachinePrecision], If[LessEqual[y, 2e+252], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.49999999999999998e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 70.3%

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

   5. Simplified 70.3%

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

   if 9.49999999999999998e-4 < y < 2.0000000000000002e252

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 82.4%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

   7. Applied egg-rr 82.4%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 70.1%

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

   if 2.0000000000000002e252 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot {y}^{4}\right) + x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \color{blue}{x} \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{{y}^{4}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(\frac{1}{120} \cdot x\right) \cdot {\color{blue}{y}}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{120}\right)}\right)\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{720} \cdot {x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. cube-unmult N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot {y}^{\color{blue}{3}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right) \cdot \color{blue}{{y}^{3}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right), \color{blue}{\left({y}^{3}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({\color{blue}{y}}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00095:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+252}:\\ \;\;\;\;\frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{if}\;x \leq 5.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)}{\frac{1 + 0.16666666666666666 \cdot \left(x \cdot x\right)}{x}}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           1.0
           (*
            (* x x)
            (+
             -0.16666666666666666
             (*
              x
              (*
               x
               (+
                0.008333333333333333
                (* (* x x) -0.0001984126984126984))))))))))
   (if (<= x 5.1e+36)
     (/
      (+
       1.0
       (*
        (* y y)
        (+
         0.16666666666666666
         (*
          (* y y)
          (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))))
      (/ (+ 1.0 (* 0.16666666666666666 (* x x))) x))
     (if (<= x 6.8e+106)
       t_0
       (if (<= x 1.45e+235)
         (*
          x
          (+
           1.0
           (*
            x
            (* x (+ -0.16666666666666666 (* 0.008333333333333333 (* x x)))))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	double tmp;
	if (x <= 5.1e+36) {
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / ((1.0 + (0.16666666666666666 * (x * x))) / x);
	} else if (x <= 6.8e+106) {
		tmp = t_0;
	} else if (x <= 1.45e+235) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + ((x * x) * (-0.0001984126984126984d0))))))))
    if (x <= 5.1d+36) then
        tmp = (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0))))))) / ((1.0d0 + (0.16666666666666666d0 * (x * x))) / x)
    else if (x <= 6.8d+106) then
        tmp = t_0
    else if (x <= 1.45d+235) then
        tmp = x * (1.0d0 + (x * (x * ((-0.16666666666666666d0) + (0.008333333333333333d0 * (x * x))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	double tmp;
	if (x <= 5.1e+36) {
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / ((1.0 + (0.16666666666666666 * (x * x))) / x);
	} else if (x <= 6.8e+106) {
		tmp = t_0;
	} else if (x <= 1.45e+235) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))))
	tmp = 0
	if x <= 5.1e+36:
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / ((1.0 + (0.16666666666666666 * (x * x))) / x)
	elif x <= 6.8e+106:
		tmp = t_0
	elif x <= 1.45e+235:
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.0001984126984126984))))))))
	tmp = 0.0
	if (x <= 5.1e+36)
		tmp = Float64(Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))))) / Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(x * x))) / x));
	elseif (x <= 6.8e+106)
		tmp = t_0;
	elseif (x <= 1.45e+235)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(0.008333333333333333 * Float64(x * x)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	tmp = 0.0;
	if (x <= 5.1e+36)
		tmp = (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984))))))) / ((1.0 + (0.16666666666666666 * (x * x))) / x);
	elseif (x <= 6.8e+106)
		tmp = t_0;
	elseif (x <= 1.45e+235)
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.1e+36], N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e+106], t$95$0, If[LessEqual[x, 1.45e+235], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.09999999999999973e36

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.7%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

   7. Applied egg-rr 92.6%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 67.5%

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

   10. Simplified 67.5%

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

   if 5.09999999999999973e36 < x < 6.79999999999999989e106 or 1.45000000000000011e235 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 47.2%

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

   8. Simplified 47.2%

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

   if 6.79999999999999989e106 < x < 1.45000000000000011e235

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 38.4%

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

   5. Simplified 38.4%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 55.5%

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

   8. Simplified 55.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)}{\frac{1 + 0.16666666666666666 \cdot \left(x \cdot x\right)}{x}}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{if}\;x \leq 5.1 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           1.0
           (*
            (* x x)
            (+
             -0.16666666666666666
             (*
              x
              (*
               x
               (+
                0.008333333333333333
                (* (* x x) -0.0001984126984126984))))))))))
   (if (<= x 5.1e+36)
     (*
      x
      (+
       1.0
       (*
        y
        (*
         y
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))
     (if (<= x 6.8e+106)
       t_0
       (if (<= x 1.45e+235)
         (*
          x
          (+
           1.0
           (*
            x
            (* x (+ -0.16666666666666666 (* 0.008333333333333333 (* x x)))))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	double tmp;
	if (x <= 5.1e+36) {
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	} else if (x <= 6.8e+106) {
		tmp = t_0;
	} else if (x <= 1.45e+235) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + ((x * x) * (-0.0001984126984126984d0))))))))
    if (x <= 5.1d+36) then
        tmp = x * (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))
    else if (x <= 6.8d+106) then
        tmp = t_0
    else if (x <= 1.45d+235) then
        tmp = x * (1.0d0 + (x * (x * ((-0.16666666666666666d0) + (0.008333333333333333d0 * (x * x))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	double tmp;
	if (x <= 5.1e+36) {
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	} else if (x <= 6.8e+106) {
		tmp = t_0;
	} else if (x <= 1.45e+235) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))))
	tmp = 0
	if x <= 5.1e+36:
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))
	elif x <= 6.8e+106:
		tmp = t_0
	elif x <= 1.45e+235:
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.0001984126984126984))))))))
	tmp = 0.0
	if (x <= 5.1e+36)
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))));
	elseif (x <= 6.8e+106)
		tmp = t_0;
	elseif (x <= 1.45e+235)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(0.008333333333333333 * Float64(x * x)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))));
	tmp = 0.0;
	if (x <= 5.1e+36)
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	elseif (x <= 6.8e+106)
		tmp = t_0;
	elseif (x <= 1.45e+235)
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.1e+36], N[(x * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e+106], t$95$0, If[LessEqual[x, 1.45e+235], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.09999999999999973e36

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.7%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 71.0%

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

   8. Simplified 71.0%

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

   if 5.09999999999999973e36 < x < 6.79999999999999989e106 or 1.45000000000000011e235 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 47.2%

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

   8. Simplified 47.2%

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

   if 6.79999999999999989e106 < x < 1.45000000000000011e235

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 38.4%

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

   5. Simplified 38.4%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 55.5%

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

   8. Simplified 55.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+106}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e+252)
   (*
    x
    (+
     1.0
     (*
      y
      (*
       y
       (+
        0.16666666666666666
        (*
         y
         (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))
   (*
    (* y (* x (+ 0.008333333333333333 (* (* x x) -0.001388888888888889))))
    (* y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e+252) {
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (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 <= 3.7d+252) then
        tmp = x * (1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))
    else
        tmp = (y * (x * (0.008333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))) * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.7e+252) {
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.7e+252:
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))
	else:
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e+252)
		tmp = Float64(x * Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.001388888888888889)))) * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.7e+252)
		tmp = x * (1.0 + (y * (y * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
	else
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e+252], N[(x * N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6999999999999998e252

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 91.9%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 62.9%

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

   8. Simplified 62.9%

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

   if 3.6999999999999998e252 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot {y}^{4}\right) + x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \color{blue}{x} \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{{y}^{4}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(\frac{1}{120} \cdot x\right) \cdot {\color{blue}{y}}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{120}\right)}\right)\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{720} \cdot {x}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. cube-unmult N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot {y}^{\color{blue}{3}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right) \cdot \color{blue}{{y}^{3}} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right), \color{blue}{\left({y}^{3}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({\color{blue}{y}}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+252}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+237}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e+18)
   (*
    x
    (+
     1.0
     (* x (* x (+ -0.16666666666666666 (* 0.008333333333333333 (* x x)))))))
   (if (<= y 1e+237)
     (* x (* y (* y (* (* y y) 0.008333333333333333))))
     (*
      (* y y)
      (* x (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+18) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else if (y <= 1e+237) {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = (y * y) * (x * (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 <= 2.3d+18) then
        tmp = x * (1.0d0 + (x * (x * ((-0.16666666666666666d0) + (0.008333333333333333d0 * (x * x))))))
    else if (y <= 1d+237) then
        tmp = x * (y * (y * ((y * y) * 0.008333333333333333d0)))
    else
        tmp = (y * y) * (x * (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 <= 2.3e+18) {
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	} else if (y <= 1e+237) {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e+18:
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))))
	elif y <= 1e+237:
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)))
	else:
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e+18)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(0.008333333333333333 * Float64(x * x)))))));
	elseif (y <= 1e+237)
		tmp = Float64(x * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))));
	else
		tmp = Float64(Float64(y * y) * Float64(x * 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 <= 2.3e+18)
		tmp = x * (1.0 + (x * (x * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))));
	elseif (y <= 1e+237)
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	else
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e+18], N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+237], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.3e18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 46.2%

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

   8. Simplified 46.2%

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

   if 2.3e18 < y < 9.9999999999999994e236

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 87.2%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. unpow3 N/A

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

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

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

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

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

      21. *-commutative N/A

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

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

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

      23. unpow2 N/A

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

      24. *-lowering-*.f64 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 73.6%

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

   if 9.9999999999999994e236 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \color{blue}{x} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(x \cdot \frac{1}{6}\right) \cdot \color{blue}{{y}^{2}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot {\color{blue}{y}}^{2} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} + \color{blue}{\frac{-1}{36} \cdot {x}^{2}}\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+237}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e+14)
   (* x (+ 1.0 (* x (* x -0.16666666666666666))))
   (if (<= y 6.2e+235)
     (* x (* y (* y (* (* y y) 0.008333333333333333))))
     (*
      (* y y)
      (* x (+ 0.16666666666666666 (* (* x x) -0.027777777777777776)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e+14) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else if (y <= 6.2e+235) {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = (y * y) * (x * (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 <= 7.5d+14) then
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    else if (y <= 6.2d+235) then
        tmp = x * (y * (y * ((y * y) * 0.008333333333333333d0)))
    else
        tmp = (y * y) * (x * (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 <= 7.5e+14) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else if (y <= 6.2e+235) {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.5e+14:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	elif y <= 6.2e+235:
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)))
	else:
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e+14)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	elseif (y <= 6.2e+235)
		tmp = Float64(x * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))));
	else
		tmp = Float64(Float64(y * y) * Float64(x * 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 <= 7.5e+14)
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	elseif (y <= 6.2e+235)
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	else
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e+14], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+235], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5e14

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 68.5%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 46.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 46.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 44.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 44.1%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right) \]

   if 7.5e14 < y < 6.20000000000000022e235

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

      14. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{3}\right)}\right)\right) \]

      16. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      24. *-lowering-*.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 87.2%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 73.6%

       \[\leadsto \color{blue}{x} \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \]

   if 6.20000000000000022e235 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \color{blue}{x} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(x \cdot \frac{1}{6}\right) \cdot \color{blue}{{y}^{2}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot {\color{blue}{y}}^{2} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} + \color{blue}{\frac{-1}{36} \cdot {x}^{2}}\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333 + 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e+255)
   (*
    x
    (+
     1.0
     (* (* y y) (+ (* (* y y) 0.008333333333333333) 0.16666666666666666))))
   (*
    (* y (* x (+ 0.008333333333333333 (* (* x x) -0.001388888888888889))))
    (* y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e+255) {
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (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 <= 1.6d+255) then
        tmp = x * (1.0d0 + ((y * y) * (((y * y) * 0.008333333333333333d0) + 0.16666666666666666d0)))
    else
        tmp = (y * (x * (0.008333333333333333d0 + ((x * x) * (-0.001388888888888889d0))))) * (y * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e+255) {
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	} else {
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e+255:
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)))
	else:
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e+255)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * 0.008333333333333333) + 0.16666666666666666))));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.001388888888888889)))) * Float64(y * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e+255)
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	else
		tmp = (y * (x * (0.008333333333333333 + ((x * x) * -0.001388888888888889)))) * (y * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e+255], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5999999999999999e255

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 62.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 62.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   if 1.5999999999999999e255 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

      14. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{3}\right)}\right)\right) \]

      16. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      24. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right) + \frac{1}{120} \cdot {y}^{4}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{720} \cdot \left({x}^{2} \cdot {y}^{4}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{720} \cdot {x}^{2}\right) \cdot {y}^{4}\right) + x \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \color{blue}{x} \cdot \left(\frac{1}{120} \cdot {y}^{4}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(x \cdot \frac{1}{120}\right) \cdot \color{blue}{{y}^{4}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)\right) \cdot {y}^{4} + \left(\frac{1}{120} \cdot x\right) \cdot {\color{blue}{y}}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{4} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{4}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)}\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{\left(2 \cdot 2\right)}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       9. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot {y}^{2}\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \left({y}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1}{120} \cdot x\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{-1}{720} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{120}}\right)\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{120}\right)}\right)\right) \]

       17. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(x \cdot \left(\frac{1}{120} + \color{blue}{\frac{-1}{720} \cdot {x}^{2}}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right) \]

       19. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

       20. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

       2. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \]

       3. cube-unmult N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot \left(y \cdot {y}^{\color{blue}{3}}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right) \cdot \color{blue}{{y}^{3}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right) \cdot y\right), \color{blue}{\left({y}^{3}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({\color{blue}{y}}^{3}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right), y\right), \left({y}^{3}\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left({y}^{3}\right)\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       13. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right), y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right) \cdot y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333 + 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.5e+127)
   (* x (+ 1.0 (* (* y y) 0.16666666666666666)))
   (if (<= x 1.45e+235)
     (* x (* 0.008333333333333333 (* (* x x) (* x x))))
     (* x (+ 1.0 (* x (* x -0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+127) {
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	} else if (x <= 1.45e+235) {
		tmp = x * (0.008333333333333333 * ((x * x) * (x * x)));
	} else {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.5d+127) then
        tmp = x * (1.0d0 + ((y * y) * 0.16666666666666666d0))
    else if (x <= 1.45d+235) then
        tmp = x * (0.008333333333333333d0 * ((x * x) * (x * x)))
    else
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+127) {
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	} else if (x <= 1.45e+235) {
		tmp = x * (0.008333333333333333 * ((x * x) * (x * x)));
	} else {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e+127:
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666))
	elif x <= 1.45e+235:
		tmp = x * (0.008333333333333333 * ((x * x) * (x * x)))
	else:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e+127)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)));
	elseif (x <= 1.45e+235)
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e+127)
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	elseif (x <= 1.45e+235)
		tmp = x * (0.008333333333333333 * ((x * x) * (x * x)));
	else
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e+127], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e+235], N[(x * N[(0.008333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.5e127

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 78.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 6.5e127 < x < 1.45000000000000011e235

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 38.4%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 38.4%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 55.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{4}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 55.5%

       \[\leadsto x \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \]

   if 1.45000000000000011e235 < x

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.3%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 7.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 7.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 47.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 47.5%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+236}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333 + 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e+236)
   (*
    x
    (+
     1.0
     (* (* y y) (+ (* (* y y) 0.008333333333333333) 0.16666666666666666))))
   (*
    (* y y)
    (* x (+ 0.16666666666666666 (* (* x x) -0.027777777777777776))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+236) {
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	} else {
		tmp = (y * y) * (x * (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 <= 4d+236) then
        tmp = x * (1.0d0 + ((y * y) * (((y * y) * 0.008333333333333333d0) + 0.16666666666666666d0)))
    else
        tmp = (y * y) * (x * (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 <= 4e+236) {
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	} else {
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+236:
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)))
	else:
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+236)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * 0.008333333333333333) + 0.16666666666666666))));
	else
		tmp = Float64(Float64(y * y) * Float64(x * 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 <= 4e+236)
		tmp = x * (1.0 + ((y * y) * (((y * y) * 0.008333333333333333) + 0.16666666666666666)));
	else
		tmp = (y * y) * (x * (0.16666666666666666 + ((x * x) * -0.027777777777777776)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+236], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.00000000000000021e236

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 61.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   if 4.00000000000000021e236 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right) + \frac{1}{6} \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto x \cdot \left(\frac{-1}{36} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) + \color{blue}{x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(\frac{-1}{36} \cdot {x}^{2}\right) \cdot {y}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \color{blue}{x} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(x \cdot \frac{1}{6}\right) \cdot \color{blue}{{y}^{2}} \]

       5. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot {\color{blue}{y}}^{2} \]

       6. distribute-rgt-out N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right)} + \frac{1}{6} \cdot x\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(\frac{-1}{36} \cdot {x}^{2}\right) + x \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

       11. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2} + \frac{1}{6}\right)}\right)\right) \]

       13. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} + \color{blue}{\frac{-1}{36} \cdot {x}^{2}}\right)\right)\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{36} \cdot {x}^{2}\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{36}}\right)\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{36}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{36}\right)\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+236}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333 + 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e+14)
   (* x (+ 1.0 (* x (* x -0.16666666666666666))))
   (* x (* y (* y (* (* y y) 0.008333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+14) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d+14) then
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    else
        tmp = x * (y * (y * ((y * y) * 0.008333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e+14) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.5e+14:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	else:
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.5e+14)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	else
		tmp = Float64(x * Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e+14)
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	else
		tmp = x * (y * (y * ((y * y) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.5e+14], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e14

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 68.5%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 46.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 46.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 44.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 44.1%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right) \]

   if 6.5e14 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \sin x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \sin x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \sin x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right)} \]

      4. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \sin x\right) + \frac{1}{6} \cdot \sin x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \sin x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \sin x + \frac{1}{6} \cdot \sin x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \sin x \cdot 1 + \left(\sin x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \sin x \cdot 1 + \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \sin x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 91.1%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot \left({y}^{4} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{120} \cdot {y}^{4}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin x \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin x \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{{y}^{2}} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120}} \cdot {y}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

      14. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(y \cdot \left(\frac{1}{120} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{3}\right)}\right)\right) \]

      16. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot \left({y}^{2} \cdot y\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      24. *-lowering-*.f64 91.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   8. Simplified 91.1%

      \[\leadsto \color{blue}{\sin x \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 71.0%

       \[\leadsto \color{blue}{x} \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e+235)
   (* x (+ 1.0 (* (* y y) 0.16666666666666666)))
   (* x (+ 1.0 (* x (* x -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+235) {
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	} else {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.45d+235) then
        tmp = x * (1.0d0 + ((y * y) * 0.16666666666666666d0))
    else
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+235) {
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	} else {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e+235:
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666))
	else:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e+235)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e+235)
		tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
	else
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e+235], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000011e235

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 77.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 77.3%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   8. Simplified 56.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 1.45000000000000011e235 < x

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.3%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 7.6%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 7.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       2. *-lowering-*.f64 47.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   11. Simplified 47.5%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+235}:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 140:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 140.0) x (* x (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 140.0) {
		tmp = x;
	} else {
		tmp = x * ((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 <= 140.0d0) then
        tmp = x
    else
        tmp = x * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 140.0) {
		tmp = x;
	} else {
		tmp = x * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 140.0:
		tmp = x
	else:
		tmp = x * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 140.0)
		tmp = x;
	else
		tmp = Float64(x * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 140.0)
		tmp = x;
	else
		tmp = x * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 140.0], x, N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 140

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 69.9%

         \[\leadsto \mathsf{sin.f64}\left(x\right) \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   7. Step-by-step derivation

   8. Simplified 40.5%

      \[\leadsto \color{blue}{x} \]

   if 140 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

      2. associate-*r* N/A

         \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      3. distribute-rgt-in N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   5. Simplified 59.7%

      \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

      2. *-commutative N/A

         \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(x \cdot {y}^{2}\right) \cdot \color{blue}{\frac{1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \frac{1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{6} \cdot \color{blue}{{y}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

       7. *-lowering-*.f64 48.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   11. Simplified 48.9%

       \[\leadsto \color{blue}{x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 140:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.7% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	return x * (1.0 + ((y * y) * 0.16666666666666666));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * 0.16666666666666666d0))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * 0.16666666666666666));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * 0.16666666666666666))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * 0.16666666666666666));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto 1 \cdot \sin x + \color{blue}{\frac{1}{6}} \cdot \left({y}^{2} \cdot \sin x\right) \]

   2. associate-*r* N/A

      \[\leadsto 1 \cdot \sin x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\sin x} \]

   3. distribute-rgt-in N/A

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 78.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

5. Simplified 78.2%

   \[\leadsto \color{blue}{\sin x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 53.2%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

8. Simplified 53.2%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

9. Final simplification 53.2%

   \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 21: 26.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\sin x} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 51.8%

      \[\leadsto \mathsf{sin.f64}\left(x\right) \]

5. Simplified 51.8%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 30.2%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))