Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 12.0s

Alternatives: 24

Speedup: 1.0×

• 49.8% 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: 95.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{if}\;x \leq 25:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (/ (sin y) y)
          (+
           1.0
           (*
            (* x x)
            (+
             0.5
             (*
              x
              (*
               x
               (+
                0.041666666666666664
                (* (* x x) 0.001388888888888889))))))))))
   (if (<= x 25.0) t_0 (if (<= x 7.2e+51) (cosh x) t_0))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	double tmp;
	if (x <= 25.0) {
		tmp = t_0;
	} else if (x <= 7.2e+51) {
		tmp = cosh(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(y) / y) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    if (x <= 25.0d0) then
        tmp = t_0
    else if (x <= 7.2d+51) then
        tmp = cosh(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	double tmp;
	if (x <= 25.0) {
		tmp = t_0;
	} else if (x <= 7.2e+51) {
		tmp = Math.cosh(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	tmp = 0
	if x <= 25.0:
		tmp = t_0
	elif x <= 7.2e+51:
		tmp = math.cosh(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))))))))
	tmp = 0.0
	if (x <= 25.0)
		tmp = t_0;
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	tmp = 0.0;
	if (x <= 25.0)
		tmp = t_0;
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 25.0], t$95$0, If[LessEqual[x, 7.2e+51], N[Cosh[x], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 25 or 7.20000000000000022e51 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 96.0%

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

   5. Simplified 96.0%

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

   if 25 < x < 7.20000000000000022e51

   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 91.7%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 91.7%

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

   7. Applied egg-rr 91.7%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+31}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 1.3e-6)
     t_0
     (if (<= x 9.8e+31)
       (cosh x)
       (if (<= x 1.35e+154)
         (* (cosh x) (+ 1.0 (* y (* y -0.16666666666666666))))
         (* t_0 (+ 1.0 (* (* x x) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 1.3e-6) {
		tmp = t_0;
	} else if (x <= 9.8e+31) {
		tmp = cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = t_0 * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 1.3d-6) then
        tmp = t_0
    else if (x <= 9.8d+31) then
        tmp = cosh(x)
    else if (x <= 1.35d+154) then
        tmp = cosh(x) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = t_0 * (1.0d0 + ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 1.3e-6) {
		tmp = t_0;
	} else if (x <= 9.8e+31) {
		tmp = Math.cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = Math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = t_0 * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 1.3e-6:
		tmp = t_0
	elif x <= 9.8e+31:
		tmp = math.cosh(x)
	elif x <= 1.35e+154:
		tmp = math.cosh(x) * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = t_0 * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 1.3e-6)
		tmp = t_0;
	elseif (x <= 9.8e+31)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 1.3e-6)
		tmp = t_0;
	elseif (x <= 9.8e+31)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = cosh(x) * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = t_0 * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 1.3e-6], t$95$0, If[LessEqual[x, 9.8e+31], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.35e+154], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.30000000000000005e-6

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 67.7%

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

   5. Simplified 67.7%

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

   if 1.30000000000000005e-6 < x < 9.79999999999999991e31

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 86.3%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 86.3%

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

   7. Applied egg-rr 86.3%

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

   if 9.79999999999999991e31 < x < 1.35000000000000003e154

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 80.0%

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

   5. Simplified 80.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2} \cdot \sin y}{y} \cdot \frac{1}{2} + \frac{\color{blue}{\sin y}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{\color{blue}{\sin y}}{y} \]

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. sin-lowering-sin.f64 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+31}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 25:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 25.0)
     (* t_0 (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664))))))
     (if (<= x 2.6e+77)
       (cosh x)
       (* t_0 (* x (* x (* (* x x) 0.041666666666666664))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 25.0) {
		tmp = t_0 * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 2.6e+77) {
		tmp = cosh(x);
	} else {
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 25.0d0) then
        tmp = t_0 * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0)))))
    else if (x <= 2.6d+77) then
        tmp = cosh(x)
    else
        tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 25.0) {
		tmp = t_0 * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 2.6e+77) {
		tmp = Math.cosh(x);
	} else {
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 25.0:
		tmp = t_0 * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))))
	elif x <= 2.6e+77:
		tmp = math.cosh(x)
	else:
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 25.0)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = Float64(t_0 * Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 25.0)
		tmp = t_0 * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 25.0], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[Cosh[x], $MachinePrecision], N[(t$95$0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 92.6%

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

   5. Simplified 92.6%

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

   if 25 < x < 2.6000000000000002e77

   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 93.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   if 2.6000000000000002e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x 1.3e-6)
     t_0
     (if (<= x 2.6e+77)
       (cosh x)
       (* t_0 (* x (* x (* (* x x) 0.041666666666666664))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= 1.3e-6) {
		tmp = t_0;
	} else if (x <= 2.6e+77) {
		tmp = cosh(x);
	} else {
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= 1.3d-6) then
        tmp = t_0
    else if (x <= 2.6d+77) then
        tmp = cosh(x)
    else
        tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= 1.3e-6) {
		tmp = t_0;
	} else if (x <= 2.6e+77) {
		tmp = Math.cosh(x);
	} else {
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= 1.3e-6:
		tmp = t_0
	elif x <= 2.6e+77:
		tmp = math.cosh(x)
	else:
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= 1.3e-6)
		tmp = t_0;
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = Float64(t_0 * Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= 1.3e-6)
		tmp = t_0;
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = t_0 * (x * (x * ((x * x) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, 1.3e-6], t$95$0, If[LessEqual[x, 2.6e+77], N[Cosh[x], $MachinePrecision], N[(t$95$0 * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.30000000000000005e-6

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 67.7%

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

   5. Simplified 67.7%

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

   if 1.30000000000000005e-6 < x < 2.6000000000000002e77

   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 88.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 2.6000000000000002e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.3e-6) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3e-6) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.3d-6) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.3e-6) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.3e-6:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3e-6)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.3e-6)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3e-6], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000005e-6

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 67.7%

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

   5. Simplified 67.7%

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

   if 1.30000000000000005e-6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 80.1%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 80.1%

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

   7. Applied egg-rr 80.1%

      \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

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 64.5%

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

   1. *-rgt-identity N/A

      \[\leadsto \cosh x \]

   2. cosh-lowering-cosh.f64 64.5%

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

7. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\cosh x} \]
8. Add Preprocessing

Alternative 8: 43.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_1 := x \cdot t\_0\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(0.125 + t\_2 \cdot \left(\left(x \cdot x\right) \cdot \left(t\_1 \cdot t\_1\right)\right)\right) \cdot \frac{1}{0.25 + t\_2 \cdot \left(t\_2 - 0.5\right)}\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))
        (t_1 (* x t_0))
        (t_2 (* x t_1)))
   (if (<= x 1.05e+39)
     (+
      1.0
      (*
       (* x x)
       (*
        (+ 0.125 (* t_2 (* (* x x) (* t_1 t_1))))
        (/ 1.0 (+ 0.25 (* t_2 (- t_2 0.5)))))))
     (if (<= x 2e+134)
       (*
        (+ 1.0 (* y (* y -0.16666666666666666)))
        (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) t_0)))))
       (* x (* x (* (* x x) 0.041666666666666664)))))))

C

double code(double x, double y) {
	double t_0 = 0.041666666666666664 + ((x * x) * 0.001388888888888889);
	double t_1 = x * t_0;
	double t_2 = x * t_1;
	double tmp;
	if (x <= 1.05e+39) {
		tmp = 1.0 + ((x * x) * ((0.125 + (t_2 * ((x * x) * (t_1 * t_1)))) * (1.0 / (0.25 + (t_2 * (t_2 - 0.5))))));
	} else if (x <= 2e+134) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * t_0))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)
    t_1 = x * t_0
    t_2 = x * t_1
    if (x <= 1.05d+39) then
        tmp = 1.0d0 + ((x * x) * ((0.125d0 + (t_2 * ((x * x) * (t_1 * t_1)))) * (1.0d0 / (0.25d0 + (t_2 * (t_2 - 0.5d0))))))
    else if (x <= 2d+134) then
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * t_0))))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.041666666666666664 + ((x * x) * 0.001388888888888889);
	double t_1 = x * t_0;
	double t_2 = x * t_1;
	double tmp;
	if (x <= 1.05e+39) {
		tmp = 1.0 + ((x * x) * ((0.125 + (t_2 * ((x * x) * (t_1 * t_1)))) * (1.0 / (0.25 + (t_2 * (t_2 - 0.5))))));
	} else if (x <= 2e+134) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * t_0))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.041666666666666664 + ((x * x) * 0.001388888888888889)
	t_1 = x * t_0
	t_2 = x * t_1
	tmp = 0
	if x <= 1.05e+39:
		tmp = 1.0 + ((x * x) * ((0.125 + (t_2 * ((x * x) * (t_1 * t_1)))) * (1.0 / (0.25 + (t_2 * (t_2 - 0.5))))))
	elif x <= 2e+134:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * t_0))))
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (x <= 1.05e+39)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.125 + Float64(t_2 * Float64(Float64(x * x) * Float64(t_1 * t_1)))) * Float64(1.0 / Float64(0.25 + Float64(t_2 * Float64(t_2 - 0.5)))))));
	elseif (x <= 2e+134)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * t_0)))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.041666666666666664 + ((x * x) * 0.001388888888888889);
	t_1 = x * t_0;
	t_2 = x * t_1;
	tmp = 0.0;
	if (x <= 1.05e+39)
		tmp = 1.0 + ((x * x) * ((0.125 + (t_2 * ((x * x) * (t_1 * t_1)))) * (1.0 / (0.25 + (t_2 * (t_2 - 0.5))))));
	elseif (x <= 2e+134)
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * t_0))));
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[x, 1.05e+39], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.125 + N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(0.25 + N[(t$95$2 * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+134], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.0499999999999999e39

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 60.9%

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

   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. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{{\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}^{3}}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}}\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}^{3}\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left({\frac{1}{2}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)}^{3}\right), \color{blue}{\left(\frac{1}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right) - \frac{1}{2} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{1}{720}\right)\right)\right)}\right)}\right)\right)\right) \]

   10. Applied egg-rr 36.4%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(0.125 + \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \frac{1}{0.25 + \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) - 0.5\right)}\right)} \]

   if 1.0499999999999999e39 < x < 1.99999999999999984e134

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 73.3%

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

   8. Simplified 73.3%

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

   if 1.99999999999999984e134 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right) \]

       10. cube-mult N/A

           \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{24}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       14. cube-mult N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

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

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

       19. *-commutative N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 80.6%

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

   11. Simplified 80.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(\left(0.125 + \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right) \cdot \frac{1}{0.25 + \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) - 0.5\right)}\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.5 - x \cdot t\_0}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))
   (if (<= x 3.5e+77)
     (+ 1.0 (/ (* (* x x) (- 0.25 (* (* x x) (* t_0 t_0)))) (- 0.5 (* x t_0))))
     (if (<= x 5e+134)
       (*
        (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))
        (+
         1.0
         (*
          y
          (*
           y
           (+
            -0.16666666666666666
            (*
             (* y y)
             (+ 0.008333333333333333 (* (* y y) -0.0001984126984126984))))))))
       (* x (* x (* (* x x) 0.041666666666666664)))))))

C

double code(double x, double y) {
	double t_0 = x * (0.041666666666666664 + ((x * x) * 0.001388888888888889));
	double tmp;
	if (x <= 3.5e+77) {
		tmp = 1.0 + (((x * x) * (0.25 - ((x * x) * (t_0 * t_0)))) / (0.5 - (x * t_0)));
	} else if (x <= 5e+134) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))
    if (x <= 3.5d+77) then
        tmp = 1.0d0 + (((x * x) * (0.25d0 - ((x * x) * (t_0 * t_0)))) / (0.5d0 - (x * t_0)))
    else if (x <= 5d+134) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * (0.008333333333333333d0 + ((y * y) * (-0.0001984126984126984d0))))))))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (0.041666666666666664 + ((x * x) * 0.001388888888888889));
	double tmp;
	if (x <= 3.5e+77) {
		tmp = 1.0 + (((x * x) * (0.25 - ((x * x) * (t_0 * t_0)))) / (0.5 - (x * t_0)));
	} else if (x <= 5e+134) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (0.041666666666666664 + ((x * x) * 0.001388888888888889))
	tmp = 0
	if x <= 3.5e+77:
		tmp = 1.0 + (((x * x) * (0.25 - ((x * x) * (t_0 * t_0)))) / (0.5 - (x * t_0)))
	elif x <= 5e+134:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))))
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))
	tmp = 0.0
	if (x <= 3.5e+77)
		tmp = Float64(1.0 + Float64(Float64(Float64(x * x) * Float64(0.25 - Float64(Float64(x * x) * Float64(t_0 * t_0)))) / Float64(0.5 - Float64(x * t_0))));
	elseif (x <= 5e+134)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * -0.0001984126984126984))))))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (0.041666666666666664 + ((x * x) * 0.001388888888888889));
	tmp = 0.0;
	if (x <= 3.5e+77)
		tmp = 1.0 + (((x * x) * (0.25 - ((x * x) * (t_0 * t_0)))) / (0.5 - (x * t_0)));
	elseif (x <= 5e+134)
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * -0.0001984126984126984)))))));
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5e+77], N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.25 - N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+134], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.5000000000000001e77

   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 62.0%

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

   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 54.0%

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

   8. Simplified 54.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 39.7%

       \[\leadsto 1 + \color{blue}{\frac{\left(0.25 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}} \]

   if 3.5000000000000001e77 < x < 4.99999999999999981e134

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 91.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 91.7%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]

   if 4.99999999999999981e134 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right) \]

       10. cube-mult N/A

           \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{24}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       14. cube-mult N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

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

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

       19. *-commutative N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 80.6%

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

   11. Simplified 80.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+77}:\\ \;\;\;\;1 + \frac{\left(x \cdot x\right) \cdot \left(0.25 - \left(x \cdot x\right) \cdot \left(\left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right) \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)}{0.5 - x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.5e+29)
   (*
    (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))
    (+
     1.0
     (* y (* y (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))
   (if (<= x 5e+134)
     (*
      (+ 1.0 (* y (* y -0.16666666666666666)))
      (+
       1.0
       (*
        (* x x)
        (+
         0.5
         (*
          (* x x)
          (+ 0.041666666666666664 (* (* x x) 0.001388888888888889)))))))
     (* x (* x (* (* x x) 0.041666666666666664))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+29) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else if (x <= 5e+134) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8.5d+29) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    else if (x <= 5d+134) then
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0))))))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+29) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else if (x <= 5e+134) {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.5e+29:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	elif x <= 5e+134:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))))
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e+29)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))));
	elseif (x <= 5e+134)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.5e+29)
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	elseif (x <= 5e+134)
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889))))));
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e+29], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+134], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.5000000000000006e29

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 55.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 55.3%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   if 8.5000000000000006e29 < x < 4.99999999999999981e134

   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 86.4%

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

   5. Simplified 86.4%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 65.2%

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

   8. Simplified 65.2%

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

   if 4.99999999999999981e134 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right) \]

       10. cube-mult N/A

           \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{24}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       14. cube-mult N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

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

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

       19. *-commutative N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 80.6%

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

   11. Simplified 80.6%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+49}:\\ \;\;\;\;\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.25e+49)
   (*
    (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664)))))
    (+
     1.0
     (* y (* y (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))
   (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) (* x (* x 0.001388888888888889))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.25e+49) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.25d+49) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0))))) * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))
    else
        tmp = 1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (x * (x * 0.001388888888888889d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.25e+49) {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	} else {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.25e+49:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))
	else:
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.25e+49)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))) * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(x * Float64(x * 0.001388888888888889))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.25e+49)
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664))))) * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))));
	else
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.25e+49], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.24999999999999991e49

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 87.9%

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

   5. Simplified 87.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 54.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 54.1%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]

   if 2.24999999999999991e49 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 79.2%

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

   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 77.3%

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

   8. Simplified 77.3%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. *-lowering-*.f64 77.3%

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

   11. Simplified 77.3%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+139)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))))))
   (*
    (+ 1.0 (* -0.16666666666666666 (* y y)))
    (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+139) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * (1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+139:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+139)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+139)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	else
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+139], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e139

   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 71.5%

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

   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 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-out N/A

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

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

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

       7. distribute-lft-out N/A

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

       8. +-commutative N/A

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

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

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

       16. unpow2 N/A

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

       17. associate-*l* N/A

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

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

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

       19. *-lowering-*.f64 64.3%

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

   11. Simplified 64.3%

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

   if 2.6999999999999998e139 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

   8. Simplified 25.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+30}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\\ \mathbf{elif}\;x \leq 10^{+133}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9e+30)
   (+
    1.0
    (* (* y y) (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))
   (if (<= x 1e+133)
     (*
      (* (* x x) (* x x))
      (+ 0.041666666666666664 (* (* y y) -0.006944444444444444)))
     (* x (* x (* (* x x) 0.041666666666666664))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9e+30) {
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	} else if (x <= 1e+133) {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9d+30) then
        tmp = 1.0d0 + ((y * y) * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))
    else if (x <= 1d+133) then
        tmp = ((x * x) * (x * x)) * (0.041666666666666664d0 + ((y * y) * (-0.006944444444444444d0)))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9e+30) {
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	} else if (x <= 1e+133) {
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9e+30:
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))
	elif x <= 1e+133:
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444))
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9e+30)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))));
	elseif (x <= 1e+133)
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(0.041666666666666664 + Float64(Float64(y * y) * -0.006944444444444444)));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9e+30)
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	elseif (x <= 1e+133)
		tmp = ((x * x) * (x * x)) * (0.041666666666666664 + ((y * y) * -0.006944444444444444));
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9e+30], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+133], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(y * y), $MachinePrecision] * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.9999999999999999e30

   1. Initial program 99.8%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 65.4%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 41.1%

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

   10. Simplified 41.1%

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

   if 8.9999999999999999e30 < x < 1e133

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right) + \frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{144} \cdot \left({y}^{2} \cdot {x}^{4}\right) + \frac{1}{24} \cdot {x}^{4} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot {x}^{4} + \color{blue}{\frac{1}{24}} \cdot {x}^{4} \]

       3. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot {y}^{2}\right) \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4} \]

       4. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right) \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4} \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24}\right) \cdot {x}^{4} + \frac{1}{24} \cdot {x}^{4} \]

       6. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24} + \frac{1}{24}\right)} \]

       7. metadata-eval N/A

          \[\leadsto {x}^{4} \cdot \left(\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24} + 1 \cdot \color{blue}{\frac{1}{24}}\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)}\right) \]

       9. +-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{1}{24} \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {y}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. distribute-rgt-in N/A

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

       19. metadata-eval N/A

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

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

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

       21. *-commutative N/A

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

   11. Simplified 56.3%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.041666666666666664 + \left(y \cdot y\right) \cdot -0.006944444444444444\right)} \]

   if 1e133 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right) \]

       10. cube-mult N/A

           \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{24}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       14. cube-mult N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

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

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

       19. *-commutative N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 80.6%

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

   11. Simplified 80.6%

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

Alternative 14: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+139)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* x (* x 0.001388888888888889))))))))
   (* (+ 1.0 (* y (* y -0.16666666666666666))) (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+139) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0)))))))
    else
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+139:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))))
	else:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+139)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889))))))));
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+139)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * (0.041666666666666664 + (x * (x * 0.001388888888888889)))))));
	else
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+139], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e139

   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 71.5%

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

   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 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-out N/A

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

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

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

       7. distribute-lft-out N/A

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

       8. +-commutative N/A

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

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

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

       16. unpow2 N/A

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

       17. associate-*l* N/A

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

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

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

       19. *-lowering-*.f64 64.3%

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

   11. Simplified 64.3%

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

   if 2.6999999999999998e139 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.2%

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

   8. Simplified 25.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+139)
   (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) (* x (* x 0.001388888888888889))))))
   (* (+ 1.0 (* y (* y -0.16666666666666666))) (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+139) then
        tmp = 1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (x * (x * 0.001388888888888889d0)))))
    else
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+139:
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))))
	else:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+139)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(x * Float64(x * 0.001388888888888889))))));
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+139)
		tmp = 1.0 + ((x * x) * (0.5 + ((x * x) * (x * (x * 0.001388888888888889)))));
	else
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+139], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e139

   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 71.5%

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

   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 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

       5. *-lowering-*.f64 64.2%

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

   11. Simplified 64.2%

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

   if 2.6999999999999998e139 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 25.2%

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

   8. Simplified 25.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.5% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+139)
   (+ 1.0 (* x (* x (* (* x x) (* (* x x) 0.001388888888888889)))))
   (* (+ 1.0 (* y (* y -0.16666666666666666))) (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * ((x * x) * ((x * x) * 0.001388888888888889))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+139) then
        tmp = 1.0d0 + (x * (x * ((x * x) * ((x * x) * 0.001388888888888889d0))))
    else
        tmp = (1.0d0 + (y * (y * (-0.16666666666666666d0)))) * (1.0d0 + ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * ((x * x) * ((x * x) * 0.001388888888888889))));
	} else {
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+139:
		tmp = 1.0 + (x * (x * ((x * x) * ((x * x) * 0.001388888888888889))))
	else:
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+139)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(Float64(x * x) * Float64(Float64(x * x) * 0.001388888888888889)))));
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+139)
		tmp = 1.0 + (x * (x * ((x * x) * ((x * x) * 0.001388888888888889))));
	else
		tmp = (1.0 + (y * (y * -0.16666666666666666))) * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+139], N[(1.0 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e139

   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 71.5%

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

   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 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-out N/A

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

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

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

       7. distribute-lft-out N/A

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

       8. +-commutative N/A

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

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

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

       16. unpow2 N/A

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

       17. associate-*l* N/A

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

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

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

       19. *-lowering-*.f64 64.3%

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

   11. Simplified 64.3%

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

   12. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 64.0%

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

   14. Simplified 64.0%

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

   if 2.6999999999999998e139 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 25.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+139)
   (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))
   (/ 1.0 (/ y (* y (+ 1.0 (* -0.16666666666666666 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+139) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    else
        tmp = 1.0d0 / (y / (y * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+139) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+139:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	else:
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+139)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))));
	else
		tmp = Float64(1.0 / Float64(y / Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+139)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	else
		tmp = 1.0 / (y / (y * (1.0 + (-0.16666666666666666 * (y * y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+139], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e139

   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 71.5%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      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 61.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 61.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 2.6999999999999998e139 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{y}{\sin y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cosh x, \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      4. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      6. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{y}{\sin y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 61.1%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sin y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 25.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   10. Simplified 25.2%

       \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+139}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+108}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+108)
   (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))
   (+
    1.0
    (* (* y y) (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+108) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else {
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d+108) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    else
        tmp = 1.0d0 + ((y * y) * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+108) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else {
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e+108:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	else:
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e+108)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))));
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e+108)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	else
		tmp = 1.0 + ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e+108], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999998e108

   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 71.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

      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 62.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   if 5.4999999999999998e108 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cosh x \cdot \frac{1}{\color{blue}{\frac{y}{\sin y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cosh x}{\color{blue}{\frac{y}{\sin y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cosh x, \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      4. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      6. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{y}{\sin y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 55.9%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sin y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 24.5%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right) \]

   10. Simplified 24.5%

       \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+43}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.05e+43)
   (+ 1.0 (* -0.16666666666666666 (* y y)))
   (* x (* x (* (* x x) 0.041666666666666664)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.05e+43) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.05d+43) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    else
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.05e+43) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.05e+43:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	else:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.05e+43)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.05e+43)
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	else
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.05e+43], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000001e43

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 59.2%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right) \]

      5. *-lowering-*.f64 36.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right) \]

   8. Simplified 36.9%

      \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666} \]

   if 1.05000000000000001e43 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      10. *-lowering-*.f64 88.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 88.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot {x}^{\left(2 \cdot 2\right)}\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\color{blue}{y}\right), y\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{sin.f64}\left(y\right)}, y\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

      12. *-lowering-*.f64 88.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   8. Simplified 88.5%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

       2. pow-sqr N/A

          \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

       4. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {x}^{2}\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto x \cdot \left(\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{24}}\right) \]

       9. unpow2 N/A

          \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right) \]

       10. cube-mult N/A

           \[\leadsto x \cdot \left({x}^{3} \cdot \frac{1}{24}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left({x}^{3} \cdot \color{blue}{\frac{1}{24}}\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{24}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot {x}^{2}\right) \cdot \frac{1}{24}\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{24}\right)}\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right) \]

       22. *-lowering-*.f64 68.2%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right) \]

   11. Simplified 68.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+43}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.8% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ 1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664))))))

C

double code(double x, double y) {
	return 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
}

Python

def code(x, y):
	return 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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 64.5%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}\right) \]

   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 55.2%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

8. Simplified 55.2%

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]
9. Add Preprocessing

Alternative 21: 39.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.46 \cdot 10^{+146}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.46e+146)
   (+ 1.0 (* -0.16666666666666666 (* y y)))
   (* (* x x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.46e+146) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.46d+146) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    else
        tmp = (x * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.46e+146) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.46e+146:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	else:
		tmp = (x * x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.46e+146)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.46e+146)
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	else
		tmp = (x * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.46e+146], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45999999999999993e146

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]
   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. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 60.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 60.9%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-1}{6}\right)\right) \]

      5. *-lowering-*.f64 34.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-1}{6}\right)\right) \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666} \]

   if 1.45999999999999993e146 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 80.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   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 80.0%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\frac{1}{2}}\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 74.0%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{0.5} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   14. Simplified 74.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.46 \cdot 10^{+146}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 29:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 29.0) 1.0 (* (* x x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 29.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 29.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 29.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 29.0:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 29.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 29.0)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 29.0], 1.0, N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 29

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 59.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 35.5%

      \[\leadsto \color{blue}{1} \]

   if 29 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cosh.f64}\left(x\right), \color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 81.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   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 63.7%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\frac{1}{2}}\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 39.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{0.5} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 39.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   14. Simplified 39.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 29:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 45.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* x x) 0.5)))

C

double code(double x, double y) {
	return 1.0 + ((x * x) * 0.5);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((x * x) * 0.5d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((x * x) * 0.5);
}

Python

def code(x, y):
	return 1.0 + ((x * x) * 0.5)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(x * x) * 0.5))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((x * x) * 0.5);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

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 64.5%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

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 45.9%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

8. Simplified 45.9%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

9. Final simplification 45.9%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 24: 26.4% 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 64.5%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 27.9%

   \[\leadsto \color{blue}{1} \]
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 2024158 
(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)))