Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.9s

Alternatives: 14

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 10^{+121}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e-16)
   (/ (sin y) y)
   (if (<= x 1e+121)
     (cosh x)
     (*
      (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))
      (+ 1.0 (* (* y y) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9e-16) {
		tmp = sin(y) / y;
	} else if (x <= 1e+121) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (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 (x <= 1.9d-16) then
        tmp = sin(y) / y
    else if (x <= 1d+121) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.9e-16) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1e+121) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.9e-16:
		tmp = math.sin(y) / y
	elif x <= 1e+121:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.9e-16)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1e+121)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.9e-16)
		tmp = sin(y) / y;
	elseif (x <= 1e+121)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9e-16], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1e+121], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.90000000000000006e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 66.3%

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

   5. Simplified 66.3%

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

   if 1.90000000000000006e-16 < x < 1.00000000000000004e121

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 88.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 88.9%

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

   7. Applied egg-rr 88.9%

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

   if 1.00000000000000004e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+121}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+121)
   (cosh x)
   (*
    (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))
    (+ 1.0 (* (* y y) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+121) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (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 (x <= 1.15d+121) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+121) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e+121:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+121)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e+121)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+121], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999e121

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 59.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 59.1%

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

   7. Applied egg-rr 59.1%

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

   if 1.1499999999999999e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 44.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_1 := x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + t\_0\right)\right)\right)\\ \mathbf{if}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_1\right)}{1 - x \cdot t\_1}\\ \mathbf{elif}\;x \leq 10^{+121}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.001388888888888889))
        (t_1 (* x (+ 0.5 (* x (* x (+ 0.041666666666666664 t_0)))))))
   (if (<= x 7e+51)
     (/ (- 1.0 (* (* x x) (* t_1 t_1))) (- 1.0 (* x t_1)))
     (if (<= x 1e+121)
       (+ 1.0 (* (* x x) (* (* x x) t_0)))
       (*
        (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))
        (+ 1.0 (* (* y y) -0.16666666666666666)))))))

C

double code(double x, double y) {
	double t_0 = (x * x) * 0.001388888888888889;
	double t_1 = x * (0.5 + (x * (x * (0.041666666666666664 + t_0))));
	double tmp;
	if (x <= 7e+51) {
		tmp = (1.0 - ((x * x) * (t_1 * t_1))) / (1.0 - (x * t_1));
	} else if (x <= 1e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * t_0));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * x) * 0.001388888888888889d0
    t_1 = x * (0.5d0 + (x * (x * (0.041666666666666664d0 + t_0))))
    if (x <= 7d+51) then
        tmp = (1.0d0 - ((x * x) * (t_1 * t_1))) / (1.0d0 - (x * t_1))
    else if (x <= 1d+121) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * t_0))
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) * 0.001388888888888889;
	double t_1 = x * (0.5 + (x * (x * (0.041666666666666664 + t_0))));
	double tmp;
	if (x <= 7e+51) {
		tmp = (1.0 - ((x * x) * (t_1 * t_1))) / (1.0 - (x * t_1));
	} else if (x <= 1e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * t_0));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * 0.001388888888888889
	t_1 = x * (0.5 + (x * (x * (0.041666666666666664 + t_0))))
	tmp = 0
	if x <= 7e+51:
		tmp = (1.0 - ((x * x) * (t_1 * t_1))) / (1.0 - (x * t_1))
	elif x <= 1e+121:
		tmp = 1.0 + ((x * x) * ((x * x) * t_0))
	else:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * 0.001388888888888889)
	t_1 = Float64(x * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + t_0)))))
	tmp = 0.0
	if (x <= 7e+51)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(t_1 * t_1))) / Float64(1.0 - Float64(x * t_1)));
	elseif (x <= 1e+121)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * t_0)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * 0.001388888888888889;
	t_1 = x * (0.5 + (x * (x * (0.041666666666666664 + t_0))));
	tmp = 0.0;
	if (x <= 7e+51)
		tmp = (1.0 - ((x * x) * (t_1 * t_1))) / (1.0 - (x * t_1));
	elseif (x <= 1e+121)
		tmp = 1.0 + ((x * x) * ((x * x) * t_0));
	else
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7e+51], N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+121], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7e51

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 57.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 48.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}{\color{blue}{1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)\right), \color{blue}{\left(1 - \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}\right) \]

   10. Applied egg-rr 31.6%

       \[\leadsto \color{blue}{\frac{1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)}{1 - x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)}} \]

   if 7e51 < x < 1.00000000000000004e121

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 85.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 85.7%

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

   11. Simplified 85.7%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   if 1.00000000000000004e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 43.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1 + \frac{1}{\frac{2 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0002314814814814815\right)\right)}{x \cdot x}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 490.0)
   (+
    1.0
    (/
     1.0
     (/
      (+
       2.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (*
          (* x x)
          (+ 0.008333333333333333 (* (* x x) -0.0002314814814814815))))))
      (* x x))))
   (if (<= x 7.5e+61)
     (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
     (if (<= x 2.1e+121)
       (+ 1.0 (* (* x x) (* (* x x) (* (* x x) 0.001388888888888889))))
       (*
        (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))
        (+ 1.0 (* (* y y) -0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0 + (1.0 / ((2.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0002314814814814815)))))) / (x * x)));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 2.1e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (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 (x <= 490.0d0) then
        tmp = 1.0d0 + (1.0d0 / ((2.0d0 + ((x * x) * ((-0.16666666666666666d0) + ((x * x) * (0.008333333333333333d0 + ((x * x) * (-0.0002314814814814815d0))))))) / (x * x)))
    else if (x <= 7.5d+61) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else if (x <= 2.1d+121) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0 + (1.0 / ((2.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0002314814814814815)))))) / (x * x)));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 2.1e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0 + (1.0 / ((2.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0002314814814814815)))))) / (x * x)))
	elif x <= 7.5e+61:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	elif x <= 2.1e+121:
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 490.0)
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.0002314814814814815)))))) / Float64(x * x))));
	elseif (x <= 7.5e+61)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	elseif (x <= 2.1e+121)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0 + (1.0 / ((2.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0002314814814814815)))))) / (x * x)));
	elseif (x <= 7.5e+61)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	elseif (x <= 2.1e+121)
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 490.0], N[(1.0 + N[(1.0 / N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0002314814814814815), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+61], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+121], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 490

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(x \cdot x\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}{\color{blue}{\frac{1}{2} \cdot \left(x \cdot x\right) - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(x \cdot x\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(x \cdot x\right)}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}{x \cdot \left(x \cdot \frac{1}{2}\right) - \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)} \cdot \left(x \cdot x\right)}\right)\right) \]

      5. fmm-def N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, \mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)\right)}{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\mathsf{fma}\left(x, x \cdot \frac{1}{2}, \mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)\right)}{\left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) - \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(x \cdot x\right)\right)}\right)}\right)\right) \]

   10. Applied egg-rr 18.2%

       \[\leadsto 1 + \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5 - \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot \left(\left(x \cdot x\right) \cdot 0.5\right) - \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{4320} \cdot {x}^{2}\right) - \frac{1}{6}\right)}{{x}^{2}}\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{4320} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   13. Simplified 30.4%

       \[\leadsto 1 + \frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0002314814814814815\right)\right)}{x \cdot x}}} \]

   if 490 < x < 7.5e61

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.3%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 46.5%

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

   13. Simplified 46.5%

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

   if 7.5e61 < x < 2.1000000000000002e121

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 92.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 92.3%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 92.3%

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

   11. Simplified 92.3%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   if 2.1000000000000002e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1 + \frac{1}{\frac{2 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0002314814814814815\right)\right)}{x \cdot x}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+121}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+121}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 520.0)
   (+
    1.0
    (*
     (* x x)
     (+
      0.5
      (* (* x x) (+ 0.041666666666666664 (* x (* x 0.001388888888888889)))))))
   (if (<= x 7.5e+61)
     (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
     (if (<= x 1.6e+121)
       (+ 1.0 (* (* x x) (* (* x x) (* (* x x) 0.001388888888888889))))
       (*
        (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) 0.041666666666666664))))
        (+ 1.0 (* (* y y) -0.16666666666666666)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.6e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (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 (x <= 520.0d0) then
        tmp = 1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))))))
    else if (x <= 7.5d+61) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else if (x <= 1.6d+121) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * 0.041666666666666664d0)))) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.6e+121) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))
	elif x <= 7.5e+61:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	elif x <= 1.6e+121:
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 520.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))))));
	elseif (x <= 7.5e+61)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	elseif (x <= 1.6e+121)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	elseif (x <= 7.5e+61)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	elseif (x <= 1.6e+121)
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * 0.041666666666666664)))) * (1.0 + ((y * y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 520.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+61], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e+121], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 520

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{720}\right)}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot \frac{1}{720}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{720}\right), \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 51.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{720}\right), x\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 51.0%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \color{blue}{\left(x \cdot 0.001388888888888889\right) \cdot x}\right)\right) \]

   if 520 < x < 7.5e61

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.3%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 46.5%

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

   13. Simplified 46.5%

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

   if 7.5e61 < x < 1.6e121

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 92.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 92.3%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 92.3%

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

   11. Simplified 92.3%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   if 1.6e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.1%

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

   5. Simplified 77.1%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 77.1%

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

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+121}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 540.0)
   (+
    1.0
    (*
     (* x x)
     (+
      0.5
      (* (* x x) (+ 0.041666666666666664 (* x (* x 0.001388888888888889)))))))
   (if (<= x 7.5e+61)
     (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
     (if (<= x 1.55e+131)
       (+ 1.0 (* (* x x) (* (* x x) (* (* x x) 0.001388888888888889))))
       (* (+ 1.0 (* (* x x) 0.5)) (+ 1.0 (* y (* y -0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 540.0) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (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 (x <= 540.0d0) then
        tmp = 1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))))))
    else if (x <= 7.5d+61) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else if (x <= 1.55d+131) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 + ((x * x) * 0.5d0)) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 540.0) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 540.0:
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))
	elif x <= 7.5e+61:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	elif x <= 1.55e+131:
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 540.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))))));
	elseif (x <= 7.5e+61)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	elseif (x <= 1.55e+131)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 540.0)
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889))))));
	elseif (x <= 7.5e+61)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	elseif (x <= 1.55e+131)
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 540.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+61], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+131], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 540

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{720}\right)}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(x \cdot \frac{1}{720}\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{720}\right), \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 51.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{720}\right), x\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 51.0%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \color{blue}{\left(x \cdot 0.001388888888888889\right) \cdot x}\right)\right) \]

   if 540 < x < 7.5e61

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.3%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 46.5%

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

   13. Simplified 46.5%

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

   if 7.5e61 < x < 1.55000000000000008e131

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 92.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 92.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 92.9%

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

   11. Simplified 92.9%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   if 1.55000000000000008e131 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. *-lowering-*.f64 73.9%

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

   11. Simplified 73.9%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 540:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 690:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 690.0)
   (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))
   (if (<= x 7.5e+61)
     (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
     (if (<= x 1.55e+131)
       (+ 1.0 (* (* x x) (* (* x x) (* (* x x) 0.001388888888888889))))
       (* (+ 1.0 (* (* x x) 0.5)) (+ 1.0 (* y (* y -0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 690.0) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (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 (x <= 690.0d0) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    else if (x <= 7.5d+61) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else if (x <= 1.55d+131) then
        tmp = 1.0d0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 + ((x * x) * 0.5d0)) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 690.0) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (x <= 7.5e+61) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 690.0:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	elif x <= 7.5e+61:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	elif x <= 1.55e+131:
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 690.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))));
	elseif (x <= 7.5e+61)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	elseif (x <= 1.55e+131)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 690.0)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	elseif (x <= 7.5e+61)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	elseif (x <= 1.55e+131)
		tmp = 1.0 + ((x * x) * ((x * x) * ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 690.0], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+61], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+131], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 690

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 55.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 51.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 51.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 50.0%

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

   11. Simplified 50.0%

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

   if 690 < x < 7.5e61

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.3%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 46.5%

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

   13. Simplified 46.5%

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

   if 7.5e61 < x < 1.55000000000000008e131

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 92.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 92.9%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 92.9%

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

   11. Simplified 92.9%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   if 1.55000000000000008e131 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. *-lowering-*.f64 73.9%

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

   11. Simplified 73.9%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 690:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq 500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))))
   (if (<= x 500.0)
     t_0
     (if (<= x 1.55e+69)
       (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
       (if (<= x 1.55e+131)
         t_0
         (*
          (+ 1.0 (* (* x x) 0.5))
          (+ 1.0 (* y (* y -0.16666666666666666)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	double tmp;
	if (x <= 500.0) {
		tmp = t_0;
	} else if (x <= 1.55e+69) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = t_0;
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    if (x <= 500.0d0) then
        tmp = t_0
    else if (x <= 1.55d+69) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else if (x <= 1.55d+131) then
        tmp = t_0
    else
        tmp = (1.0d0 + ((x * x) * 0.5d0)) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	double tmp;
	if (x <= 500.0) {
		tmp = t_0;
	} else if (x <= 1.55e+69) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else if (x <= 1.55e+131) {
		tmp = t_0;
	} else {
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	tmp = 0
	if x <= 500.0:
		tmp = t_0
	elif x <= 1.55e+69:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	elif x <= 1.55e+131:
		tmp = t_0
	else:
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))
	tmp = 0.0
	if (x <= 500.0)
		tmp = t_0;
	elseif (x <= 1.55e+69)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	elseif (x <= 1.55e+131)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	tmp = 0.0;
	if (x <= 500.0)
		tmp = t_0;
	elseif (x <= 1.55e+69)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	elseif (x <= 1.55e+131)
		tmp = t_0;
	else
		tmp = (1.0 + ((x * x) * 0.5)) * (1.0 + (y * (y * -0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 500.0], t$95$0, If[LessEqual[x, 1.55e+69], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+131], t$95$0, N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 500 or 1.5499999999999999e69 < x < 1.55000000000000008e131

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 57.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 53.6%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 52.2%

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

   11. Simplified 52.2%

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

   if 500 < x < 1.5499999999999999e69

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.2%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.6%

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

   13. Simplified 42.6%

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

   if 1.55000000000000008e131 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

      13. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   9. Taylor expanded in x around 0

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

       1. associate-+r+ N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. *-lowering-*.f64 73.9%

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

   11. Simplified 73.9%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+131}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot 0.5\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;x \leq 320:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))))
   (if (<= x 320.0)
     t_0
     (if (<= x 1.55e+69)
       (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	double tmp;
	if (x <= 320.0) {
		tmp = t_0;
	} else if (x <= 1.55e+69) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    if (x <= 320.0d0) then
        tmp = t_0
    else if (x <= 1.55d+69) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	double tmp;
	if (x <= 320.0) {
		tmp = t_0;
	} else if (x <= 1.55e+69) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	tmp = 0
	if x <= 320.0:
		tmp = t_0
	elif x <= 1.55e+69:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))
	tmp = 0.0
	if (x <= 320.0)
		tmp = t_0;
	elseif (x <= 1.55e+69)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	tmp = 0.0;
	if (x <= 320.0)
		tmp = t_0;
	elseif (x <= 1.55e+69)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 320.0], t$95$0, If[LessEqual[x, 1.55e+69], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 320 or 1.5499999999999999e69 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 58.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2}} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 55.1%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 53.9%

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

   11. Simplified 53.9%

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

   if 320 < x < 1.5499999999999999e69

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 41.7%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.8%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 1.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 1.2%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 42.6%

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

   13. Simplified 42.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 320:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+69}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;x \leq 650:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+151}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) 0.5))))
   (if (<= x 650.0)
     t_0
     (if (<= x 7e+151)
       (* y (* y (+ -0.16666666666666666 (/ 1.0 (* y y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x * x) * 0.5);
	double tmp;
	if (x <= 650.0) {
		tmp = t_0;
	} else if (x <= 7e+151) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * x) * 0.5d0)
    if (x <= 650.0d0) then
        tmp = t_0
    else if (x <= 7d+151) then
        tmp = y * (y * ((-0.16666666666666666d0) + (1.0d0 / (y * y))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x * x) * 0.5);
	double tmp;
	if (x <= 650.0) {
		tmp = t_0;
	} else if (x <= 7e+151) {
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x * x) * 0.5)
	tmp = 0
	if x <= 650.0:
		tmp = t_0
	elif x <= 7e+151:
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * 0.5))
	tmp = 0.0
	if (x <= 650.0)
		tmp = t_0;
	elseif (x <= 7e+151)
		tmp = Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(1.0 / Float64(y * y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x * x) * 0.5);
	tmp = 0.0;
	if (x <= 650.0)
		tmp = t_0;
	elseif (x <= 7e+151)
		tmp = y * (y * (-0.16666666666666666 + (1.0 / (y * y))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 650.0], t$95$0, If[LessEqual[x, 7e+151], N[(y * N[(y * N[(-0.16666666666666666 + N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 650 or 7.0000000000000006e151 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 58.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 47.1%

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

   8. Simplified 47.1%

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

   if 650 < x < 7.0000000000000006e151

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 1.9%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 10.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 10.3%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 35.8%

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

   13. Simplified 35.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 650:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+151}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-0.16666666666666666 + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+170}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.52e+170)
   (+ 1.0 (* (* x x) 0.5))
   (* y (* y -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.52e+170) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = y * (y * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.52d+170) then
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    else
        tmp = y * (y * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.52e+170) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = y * (y * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.52e+170:
		tmp = 1.0 + ((x * x) * 0.5)
	else:
		tmp = y * (y * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.52e+170)
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(y * Float64(y * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.52e+170)
		tmp = 1.0 + ((x * x) * 0.5);
	else
		tmp = y * (y * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.52e+170], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.51999999999999997e170

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 66.2%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 45.8%

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

   8. Simplified 45.8%

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

   if 1.51999999999999997e170 < y

   1. Initial program 99.6%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 3.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 4.2%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 0.0%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 30.2%

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

   13. Simplified 30.2%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+170}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.8% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e+82) 1.0 (* y (* y -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+82) {
		tmp = 1.0;
	} else {
		tmp = y * (y * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.3d+82) then
        tmp = 1.0d0
    else
        tmp = y * (y * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e+82) {
		tmp = 1.0;
	} else {
		tmp = y * (y * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e+82:
		tmp = 1.0
	else:
		tmp = y * (y * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e+82)
		tmp = 1.0;
	else
		tmp = Float64(y * Float64(y * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e+82)
		tmp = 1.0;
	else
		tmp = y * (y * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e+82], 1.0, N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999988e82

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 71.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 30.6%

      \[\leadsto \color{blue}{1} \]

   if 2.29999999999999988e82 < y

   1. Initial program 99.8%

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

      1. associate-*r/ N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\color{blue}{\mathsf{neg}\left(y\right)}} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 - \color{blue}{y}} \]

      4. flip-- N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{\color{blue}{0 + y}}} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{\frac{0 \cdot 0 - y \cdot y}{y}} \]

      6. associate-/r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y} \cdot \color{blue}{y} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\cosh x \cdot \sin y\right)}{0 \cdot 0 - y \cdot y}\right), \color{blue}{y}\right) \]

   4. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{\frac{-1}{\cosh x}}}{0 - y \cdot y} \cdot y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \color{blue}{-1}\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   6. Step-by-step derivation

   7. Simplified 18.5%

      \[\leadsto \frac{\frac{\sin y}{\color{blue}{-1}}}{0 - y \cdot y} \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right)}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} - 1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot {y}^{2} + -1\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left({y}^{2} \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

      10. *-lowering-*.f64 9.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right) \]

   10. Simplified 9.2%

       \[\leadsto \frac{\color{blue}{y \cdot \left(-1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)}}{0 - y \cdot y} \cdot y \]

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 18.9%

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

   13. Simplified 18.9%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

5. Simplified 60.0%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 24.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))