Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 21

Speedup: 1.0×

• 49.4% 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: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{\frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 40.0)
   (*
    (/ (sin y) y)
    (+
     1.0
     (*
      (* x x)
      (+
       0.5
       (*
        x
        (* x (+ 0.041666666666666664 (* (* x x) 0.001388888888888889))))))))
   (if (<= x 7.2e+51)
     (cosh x)
     (/
      (+ 1.0 (* (* x x) (+ 0.5 (* x (* x (* x (* x 0.001388888888888889)))))))
      (/ y (sin y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 40.0) {
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	} else if (x <= 7.2e+51) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / sin(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 40.0d0) then
        tmp = (sin(y) / y) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))))
    else if (x <= 7.2d+51) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (x * (x * 0.001388888888888889d0))))))) / (y / sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 40.0) {
		tmp = (Math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	} else if (x <= 7.2e+51) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 40.0:
		tmp = (math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))))
	elif x <= 7.2e+51:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / math.sin(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 40.0)
		tmp = 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))))))));
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(x * Float64(x * 0.001388888888888889))))))) / Float64(y / sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 40.0)
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))));
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 40.0], 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, 7.2e+51], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 40

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 95.8%

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

   5. Simplified 95.8%

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

   if 40 < 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 100.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 7.20000000000000022e51 < x

   1. Initial program 100.0%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}\right)}{\frac{y}{\sin y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{\frac{y}{\sin y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{\frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.026)
   (*
    (/ (sin y) y)
    (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664))))))
   (if (<= x 7.2e+51)
     (cosh x)
     (/
      (+ 1.0 (* (* x x) (+ 0.5 (* x (* x (* x (* x 0.001388888888888889)))))))
      (/ y (sin y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 7.2e+51) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / sin(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.026d0) then
        tmp = (sin(y) / y) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0)))))
    else if (x <= 7.2d+51) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((x * x) * (0.5d0 + (x * (x * (x * (x * 0.001388888888888889d0))))))) / (y / sin(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = (Math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 7.2e+51) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / Math.sin(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.026:
		tmp = (math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))))
	elif x <= 7.2e+51:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / math.sin(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.026)
		tmp = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * Float64(x * Float64(x * 0.001388888888888889))))))) / Float64(y / sin(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.026)
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	elseif (x <= 7.2e+51)
		tmp = cosh(x);
	else
		tmp = (1.0 + ((x * x) * (0.5 + (x * (x * (x * (x * 0.001388888888888889))))))) / (y / sin(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.026], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * 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, 7.2e+51], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0259999999999999988

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 94.1%

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

   5. Simplified 94.1%

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

   if 0.0259999999999999988 < x < 7.20000000000000022e51

   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 93.9%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 93.9%

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

   7. Applied egg-rr 93.9%

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

   if 7.20000000000000022e51 < x

   1. Initial program 100.0%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)}\right)}{\frac{y}{\sin y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)}{\frac{y}{\sin y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.026)
   (*
    (/ (sin y) y)
    (+ 1.0 (* (* x x) (+ 0.5 (* x (* x 0.041666666666666664))))))
   (if (<= x 2e+77)
     (cosh x)
     (/ (* (sin y) (* x (* x (* (* x x) 0.041666666666666664)))) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 2e+77) {
		tmp = cosh(x);
	} else {
		tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.026d0) then
        tmp = (sin(y) / y) * (1.0d0 + ((x * x) * (0.5d0 + (x * (x * 0.041666666666666664d0)))))
    else if (x <= 2d+77) then
        tmp = cosh(x)
    else
        tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664d0)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = (Math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	} else if (x <= 2e+77) {
		tmp = Math.cosh(x);
	} else {
		tmp = (Math.sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.026:
		tmp = (math.sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))))
	elif x <= 2e+77:
		tmp = math.cosh(x)
	else:
		tmp = (math.sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.026)
		tmp = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664))))));
	elseif (x <= 2e+77)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(sin(y) * Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.026)
		tmp = (sin(y) / y) * (1.0 + ((x * x) * (0.5 + (x * (x * 0.041666666666666664)))));
	elseif (x <= 2e+77)
		tmp = cosh(x);
	else
		tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.026], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * 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, 2e+77], N[Cosh[x], $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0259999999999999988

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 94.1%

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

   5. Simplified 94.1%

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

   if 0.0259999999999999988 < x < 1.99999999999999997e77

   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 91.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 91.8%

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

   7. Applied egg-rr 91.8%

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

   if 1.99999999999999997e77 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0135:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0135)
   (* (/ (sin y) y) (+ 1.0 (* (* x x) 0.5)))
   (if (<= x 2.6e+77)
     (cosh x)
     (/ (* (sin y) (* x (* x (* (* x x) 0.041666666666666664)))) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0135) {
		tmp = (sin(y) / y) * (1.0 + ((x * x) * 0.5));
	} else if (x <= 2.6e+77) {
		tmp = cosh(x);
	} else {
		tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0135d0) then
        tmp = (sin(y) / y) * (1.0d0 + ((x * x) * 0.5d0))
    else if (x <= 2.6d+77) then
        tmp = cosh(x)
    else
        tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664d0)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0135) {
		tmp = (Math.sin(y) / y) * (1.0 + ((x * x) * 0.5));
	} else if (x <= 2.6e+77) {
		tmp = Math.cosh(x);
	} else {
		tmp = (Math.sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0135:
		tmp = (math.sin(y) / y) * (1.0 + ((x * x) * 0.5))
	elif x <= 2.6e+77:
		tmp = math.cosh(x)
	else:
		tmp = (math.sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0135)
		tmp = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(sin(y) * Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0135)
		tmp = (sin(y) / y) * (1.0 + ((x * x) * 0.5));
	elseif (x <= 2.6e+77)
		tmp = cosh(x);
	else
		tmp = (sin(y) * (x * (x * ((x * x) * 0.041666666666666664)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0135], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[Cosh[x], $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0134999999999999998

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{\sin y}{y}} \]
   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 88.2%

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

   5. Simplified 88.2%

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

   if 0.0134999999999999998 < 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 91.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 91.8%

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

   7. Applied egg-rr 91.8%

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

   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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0135:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{if}\;x \leq 0.021:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+152}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) (+ 1.0 (* (* x x) 0.5)))))
   (if (<= x 0.021) t_0 (if (<= x 4.4e+152) (cosh x) t_0))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * (1.0 + ((x * x) * 0.5));
	double tmp;
	if (x <= 0.021) {
		tmp = t_0;
	} else if (x <= 4.4e+152) {
		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))
    if (x <= 0.021d0) then
        tmp = t_0
    else if (x <= 4.4d+152) 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));
	double tmp;
	if (x <= 0.021) {
		tmp = t_0;
	} else if (x <= 4.4e+152) {
		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))
	tmp = 0
	if x <= 0.021:
		tmp = t_0
	elif x <= 4.4e+152:
		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) * 0.5)))
	tmp = 0.0
	if (x <= 0.021)
		tmp = t_0;
	elseif (x <= 4.4e+152)
		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));
	tmp = 0.0;
	if (x <= 0.021)
		tmp = t_0;
	elseif (x <= 4.4e+152)
		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] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.021], t$95$0, If[LessEqual[x, 4.4e+152], N[Cosh[x], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0210000000000000013 or 4.3999999999999996e152 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot \sin y}{y} + \frac{\sin y}{y}} \]
   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 89.6%

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

   5. Simplified 89.6%

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

   if 0.0210000000000000013 < x < 4.3999999999999996e152

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.2%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 92.2%

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

   7. Applied egg-rr 92.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.021:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+152}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+152}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.5e-5)
   (/ (sin y) y)
   (if (<= x 4.4e+152) (cosh x) (* (sin y) (/ (* (* x x) 0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.5e-5) {
		tmp = sin(y) / y;
	} else if (x <= 4.4e+152) {
		tmp = cosh(x);
	} else {
		tmp = sin(y) * (((x * x) * 0.5) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.5d-5) then
        tmp = sin(y) / y
    else if (x <= 4.4d+152) then
        tmp = cosh(x)
    else
        tmp = sin(y) * (((x * x) * 0.5d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.5e-5) {
		tmp = Math.sin(y) / y;
	} else if (x <= 4.4e+152) {
		tmp = Math.cosh(x);
	} else {
		tmp = Math.sin(y) * (((x * x) * 0.5) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.5e-5:
		tmp = math.sin(y) / y
	elif x <= 4.4e+152:
		tmp = math.cosh(x)
	else:
		tmp = math.sin(y) * (((x * x) * 0.5) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.5e-5)
		tmp = Float64(sin(y) / y);
	elseif (x <= 4.4e+152)
		tmp = cosh(x);
	else
		tmp = Float64(sin(y) * Float64(Float64(Float64(x * x) * 0.5) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.5e-5)
		tmp = sin(y) / y;
	elseif (x <= 4.4e+152)
		tmp = cosh(x);
	else
		tmp = sin(y) * (((x * x) * 0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.5e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4.4e+152], N[Cosh[x], $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.50000000000000012e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 72.4%

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

   5. Simplified 72.4%

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

   if 2.50000000000000012e-5 < x < 4.3999999999999996e152

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 92.2%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 92.2%

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

   7. Applied egg-rr 92.2%

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

   if 4.3999999999999996e152 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+152}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\left(x \cdot x\right) \cdot 0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+174}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e-5)
   (/ (sin y) y)
   (if (<= x 1.5e+174)
     (cosh x)
     (*
      (* x x)
      (*
       (* x x)
       (+ 0.041666666666666664 (* -0.006944444444444444 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e-5) {
		tmp = sin(y) / y;
	} else if (x <= 1.5e+174) {
		tmp = cosh(x);
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (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 (x <= 1.75d-5) then
        tmp = sin(y) / y
    else if (x <= 1.5d+174) then
        tmp = cosh(x)
    else
        tmp = (x * x) * ((x * x) * (0.041666666666666664d0 + ((-0.006944444444444444d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.75e-5) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1.5e+174) {
		tmp = Math.cosh(x);
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.75e-5:
		tmp = math.sin(y) / y
	elif x <= 1.5e+174:
		tmp = math.cosh(x)
	else:
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e-5)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1.5e+174)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.006944444444444444 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.75e-5)
		tmp = sin(y) / y;
	elseif (x <= 1.5e+174)
		tmp = cosh(x);
	else
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e-5], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.5e+174], N[Cosh[x], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.006944444444444444 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.7499999999999998e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. sin-lowering-sin.f64 72.4%

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

   5. Simplified 72.4%

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

   if 1.7499999999999998e-5 < x < 1.5e174

   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 90.8%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 90.8%

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

   7. Applied egg-rr 90.8%

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

   if 1.5e174 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(y \cdot \frac{-1}{144}\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{144} \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot x\right)\right) \]

       12. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\left(\left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+174}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+175}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e+175)
   (cosh x)
   (*
    (* x x)
    (* (* x x) (+ 0.041666666666666664 (* -0.006944444444444444 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e+175) {
		tmp = cosh(x);
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (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 (x <= 6d+175) then
        tmp = cosh(x)
    else
        tmp = (x * x) * ((x * x) * (0.041666666666666664d0 + ((-0.006944444444444444d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6e+175) {
		tmp = Math.cosh(x);
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e+175:
		tmp = math.cosh(x)
	else:
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e+175)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.006944444444444444 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e+175)
		tmp = cosh(x);
	else
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e+175], N[Cosh[x], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.006944444444444444 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000003e175

   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 69.1%

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

      1. *-rgt-identity N/A

         \[\leadsto \cosh x \]

      2. cosh-lowering-cosh.f64 69.1%

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

   7. Applied egg-rr 69.1%

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

   if 6.0000000000000003e175 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(y \cdot \frac{-1}{144}\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{144} \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot x\right)\right) \]

       12. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\left(\left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+175}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}{\frac{y}{y \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+174)
   (/
    (+
     1.0
     (*
      (* x x)
      (+
       0.5
       (* (* x x) (+ 0.041666666666666664 (* (* x x) 0.001388888888888889))))))
    (/
     y
     (*
      y
      (+
       1.0
       (*
        y
        (* y (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))))
   (*
    (* x x)
    (* (* x x) (+ 0.041666666666666664 (* -0.006944444444444444 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+174) {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))) / (y / (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (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 (x <= 2d+174) then
        tmp = (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (0.041666666666666664d0 + ((x * x) * 0.001388888888888889d0)))))) / (y / (y * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))))
    else
        tmp = (x * x) * ((x * x) * (0.041666666666666664d0 + ((-0.006944444444444444d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e+174) {
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))) / (y / (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e+174:
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))) / (y / (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))))
	else:
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+174)
		tmp = Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889)))))) / Float64(y / Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))))));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.006944444444444444 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2e+174)
		tmp = (1.0 + ((x * x) * (0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * 0.001388888888888889)))))) / (y / (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))));
	else
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+174], N[(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] / N[(y / N[(y * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.006944444444444444 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000014e174

   1. Initial program 99.9%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {y}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 63.9%

       \[\leadsto \frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}{\frac{y}{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}} \]

   if 2.00000000000000014e174 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(y \cdot \frac{-1}{144}\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{144} \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot x\right)\right) \]

       12. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\left(\left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+174}:\\ \;\;\;\;\frac{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\right)}{\frac{y}{y \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right) \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+176}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e+51)
   (*
    (*
     y
     (+
      1.0
      (* y (* y (+ -0.16666666666666666 (* (* y y) 0.008333333333333333))))))
    (/ (+ 1.0 (* (* x x) 0.5)) y))
   (if (<= x 8e+176)
     (+
      1.0
      (*
       x
       (*
        x
        (+
         0.5
         (*
          x
          (* x (+ 0.041666666666666664 (* x (* x 0.001388888888888889)))))))))
     (*
      (* x x)
      (*
       (* x x)
       (+ 0.041666666666666664 (* -0.006944444444444444 (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+51) {
		tmp = (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (x <= 8e+176) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (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 (x <= 7.2d+51) then
        tmp = (y * (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0)))))) * ((1.0d0 + ((x * x) * 0.5d0)) / y)
    else if (x <= 8d+176) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + (x * (x * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))))))))
    else
        tmp = (x * x) * ((x * x) * (0.041666666666666664d0 + ((-0.006944444444444444d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+51) {
		tmp = (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (x <= 8e+176) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e+51:
		tmp = (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) * ((1.0 + ((x * x) * 0.5)) / y)
	elif x <= 8e+176:
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))))
	else:
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e+51)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))) * Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) / y));
	elseif (x <= 8e+176)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))))))));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.006944444444444444 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e+51)
		tmp = (y * (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * 0.008333333333333333)))))) * ((1.0 + ((x * x) * 0.5)) / y);
	elseif (x <= 8e+176)
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	else
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e+51], N[(N[(y * N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+176], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.006944444444444444 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.20000000000000022e51

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 59.2%

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

   10. Simplified 59.2%

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

   if 7.20000000000000022e51 < x < 8.0000000000000001e176

   1. Initial program 100.0%

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 88.9%

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

   10. Simplified 88.9%

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

   if 8.0000000000000001e176 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(y \cdot \frac{-1}{144}\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{144} \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot x\right)\right) \]

       12. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\left(\left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right) \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+176}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.05 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;x \leq 10^{+230}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.041666666666666664 \cdot t\_0\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\_0\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x 1.05e+21)
     (* y (/ (+ 1.0 (* (* x x) 0.5)) y))
     (if (<= x 1e+230)
       (/ (* y (* x (* 0.041666666666666664 t_0))) y)
       (*
        (* x t_0)
        (+ 0.041666666666666664 (* y (* y -0.006944444444444444))))))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.05e+21) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (x <= 1e+230) {
		tmp = (y * (x * (0.041666666666666664 * t_0))) / y;
	} else {
		tmp = (x * t_0) * (0.041666666666666664 + (y * (y * -0.006944444444444444)));
	}
	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 * (x * x)
    if (x <= 1.05d+21) then
        tmp = y * ((1.0d0 + ((x * x) * 0.5d0)) / y)
    else if (x <= 1d+230) then
        tmp = (y * (x * (0.041666666666666664d0 * t_0))) / y
    else
        tmp = (x * t_0) * (0.041666666666666664d0 + (y * (y * (-0.006944444444444444d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= 1.05e+21) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (x <= 1e+230) {
		tmp = (y * (x * (0.041666666666666664 * t_0))) / y;
	} else {
		tmp = (x * t_0) * (0.041666666666666664 + (y * (y * -0.006944444444444444)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * x)
	tmp = 0
	if x <= 1.05e+21:
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y)
	elif x <= 1e+230:
		tmp = (y * (x * (0.041666666666666664 * t_0))) / y
	else:
		tmp = (x * t_0) * (0.041666666666666664 + (y * (y * -0.006944444444444444)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= 1.05e+21)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) / y));
	elseif (x <= 1e+230)
		tmp = Float64(Float64(y * Float64(x * Float64(0.041666666666666664 * t_0))) / y);
	else
		tmp = Float64(Float64(x * t_0) * Float64(0.041666666666666664 + Float64(y * Float64(y * -0.006944444444444444))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= 1.05e+21)
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	elseif (x <= 1e+230)
		tmp = (y * (x * (0.041666666666666664 * t_0))) / y;
	else
		tmp = (x * t_0) * (0.041666666666666664 + (y * (y * -0.006944444444444444)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.05e+21], N[(y * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+230], N[(N[(y * N[(x * N[(0.041666666666666664 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] * N[(0.041666666666666664 + N[(y * N[(y * -0.006944444444444444), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.05e21

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 55.3%

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

   if 1.05e21 < x < 1.0000000000000001e230

   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 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot y\right), y\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{1}{24} \cdot {x}^{4}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{24} \cdot {x}^{4}\right)\right), y\right) \]

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*r* N/A

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

       12. unpow2 N/A

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

       13. unpow3 N/A

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

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

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 66.4%

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

   11. Simplified 66.4%

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

   if 1.0000000000000001e230 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 95.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 95.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;x \leq 10^{+230}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+175}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e+175)
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.5
       (*
        x
        (* x (+ 0.041666666666666664 (* x (* x 0.001388888888888889)))))))))
   (*
    (* x x)
    (* (* x x) (+ 0.041666666666666664 (* -0.006944444444444444 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+175) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (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 (x <= 1d+175) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + (x * (x * (0.041666666666666664d0 + (x * (x * 0.001388888888888889d0))))))))
    else
        tmp = (x * x) * ((x * x) * (0.041666666666666664d0 + ((-0.006944444444444444d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+175) {
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	} else {
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+175:
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))))
	else:
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+175)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(x * Float64(x * 0.001388888888888889)))))))));
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.006944444444444444 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+175)
		tmp = 1.0 + (x * (x * (0.5 + (x * (x * (0.041666666666666664 + (x * (x * 0.001388888888888889))))))));
	else
		tmp = (x * x) * ((x * x) * (0.041666666666666664 + (-0.006944444444444444 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+175], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(x * N[(x * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.006944444444444444 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e174

   1. Initial program 99.9%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} \]
   9. 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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-lowering-*.f64 60.3%

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

   10. Simplified 60.3%

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

   if 9.9999999999999994e174 < 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

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 92.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(x \cdot x\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{24} + y \cdot \left(y \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(\color{blue}{x} \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(y \cdot \frac{-1}{144}\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{144} \cdot \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \left(y \cdot y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(x \cdot x\right)\right), \left(x \cdot x\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x \cdot x\right)\right) \]

       12. *-lowering-*.f64 92.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\frac{-1}{144}, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\left(\left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+175}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.006944444444444444 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{+158}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.95e+90)
   (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))
   (if (<= y 2.06e+158)
     (* (* y (* y -0.006944444444444444)) (* (* x x) (* x x)))
     (/ (* y (* x (* 0.041666666666666664 (* x (* x x))))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (y <= 2.06e+158) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = (y * (x * (0.041666666666666664 * (x * (x * x))))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.95d+90) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    else if (y <= 2.06d+158) then
        tmp = (y * (y * (-0.006944444444444444d0))) * ((x * x) * (x * x))
    else
        tmp = (y * (x * (0.041666666666666664d0 * (x * (x * x))))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (y <= 2.06e+158) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = (y * (x * (0.041666666666666664 * (x * (x * x))))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.95e+90:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	elif y <= 2.06e+158:
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x))
	else:
		tmp = (y * (x * (0.041666666666666664 * (x * (x * x))))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.95e+90)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))));
	elseif (y <= 2.06e+158)
		tmp = Float64(Float64(y * Float64(y * -0.006944444444444444)) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.041666666666666664 * Float64(x * Float64(x * x))))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.95e+90)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	elseif (y <= 2.06e+158)
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	else
		tmp = (y * (x * (0.041666666666666664 * (x * (x * x))))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.95e+90], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.06e+158], N[(N[(y * N[(y * -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.9499999999999998e90

   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 76.5%

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

   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 65.0%

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

   8. Simplified 65.0%

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

   if 3.9499999999999998e90 < y < 2.06000000000000004e158

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 55.7%

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

   8. Simplified 55.7%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 40.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 40.9%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{144}} \]

       2. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{144}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{-1}{144} \cdot \color{blue}{{y}^{2}}\right) \]

       4. metadata-eval N/A

          \[\leadsto {x}^{4} \cdot \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot {\color{blue}{y}}^{2}\right) \]

       5. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. associate-*r* N/A

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

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{-1}{144} \cdot {\color{blue}{y}}^{2}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right) \]

       21. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       22. *-lowering-*.f64 40.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

   14. Simplified 40.9%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   if 2.06000000000000004e158 < y

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 25.2%

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

   8. Simplified 25.2%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot y\right), y\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(\frac{1}{24} \cdot {x}^{4}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{1}{24} \cdot {x}^{4}\right)\right), y\right) \]

       4. metadata-eval N/A

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

       5. pow-sqr N/A

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

       6. associate-*l* N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*r* N/A

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

       12. unpow2 N/A

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

       13. unpow3 N/A

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

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

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 24.7%

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

   11. Simplified 24.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;y \leq 2.06 \cdot 10^{+158}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.95e+90)
   (+ 1.0 (* x (* x (+ 0.5 (* (* x x) 0.041666666666666664)))))
   (if (<= y 1.15e+203)
     (* (* y (* y -0.006944444444444444)) (* (* x x) (* x x)))
     (* x (* 0.041666666666666664 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (y <= 1.15e+203) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.95d+90) then
        tmp = 1.0d0 + (x * (x * (0.5d0 + ((x * x) * 0.041666666666666664d0))))
    else if (y <= 1.15d+203) then
        tmp = (y * (y * (-0.006944444444444444d0))) * ((x * x) * (x * x))
    else
        tmp = x * (0.041666666666666664d0 * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	} else if (y <= 1.15e+203) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.95e+90:
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))))
	elif y <= 1.15e+203:
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x))
	else:
		tmp = x * (0.041666666666666664 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.95e+90)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))));
	elseif (y <= 1.15e+203)
		tmp = Float64(Float64(y * Float64(y * -0.006944444444444444)) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.95e+90)
		tmp = 1.0 + (x * (x * (0.5 + ((x * x) * 0.041666666666666664))));
	elseif (y <= 1.15e+203)
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	else
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.95e+90], N[(1.0 + N[(x * N[(x * N[(0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+203], N[(N[(y * N[(y * -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.9499999999999998e90

   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 76.5%

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

   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 65.0%

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

   8. Simplified 65.0%

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

   if 3.9499999999999998e90 < y < 1.15e203

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sin-lowering-sin.f64 40.7%

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

   8. Simplified 40.7%

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

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 34.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 34.8%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{144}} \]

       2. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{144}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{-1}{144} \cdot \color{blue}{{y}^{2}}\right) \]

       4. metadata-eval N/A

          \[\leadsto {x}^{4} \cdot \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot {\color{blue}{y}}^{2}\right) \]

       5. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. associate-*r* N/A

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

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{-1}{144} \cdot {\color{blue}{y}}^{2}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right) \]

       21. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       22. *-lowering-*.f64 34.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

   14. Simplified 34.8%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   if 1.15e203 < y

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 16.8%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 19.1%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 19.1%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.95e+90)
   (* y (/ (+ 1.0 (* (* x x) 0.5)) y))
   (if (<= y 1.15e+203)
     (* (* y (* y -0.006944444444444444)) (* (* x x) (* x x)))
     (* x (* 0.041666666666666664 (* x (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (y <= 1.15e+203) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.95d+90) then
        tmp = y * ((1.0d0 + ((x * x) * 0.5d0)) / y)
    else if (y <= 1.15d+203) then
        tmp = (y * (y * (-0.006944444444444444d0))) * ((x * x) * (x * x))
    else
        tmp = x * (0.041666666666666664d0 * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.95e+90) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else if (y <= 1.15e+203) {
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.95e+90:
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y)
	elif y <= 1.15e+203:
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x))
	else:
		tmp = x * (0.041666666666666664 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.95e+90)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) / y));
	elseif (y <= 1.15e+203)
		tmp = Float64(Float64(y * Float64(y * -0.006944444444444444)) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.95e+90)
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	elseif (y <= 1.15e+203)
		tmp = (y * (y * -0.006944444444444444)) * ((x * x) * (x * x));
	else
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.95e+90], N[(y * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+203], N[(N[(y * N[(y * -0.006944444444444444), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.9499999999999998e90

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. associate-/r/ N/A

         \[\leadsto \cosh x \cdot \left(\frac{1}{y} \cdot \color{blue}{\sin y}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\cosh x \cdot \frac{1}{y}\right) \cdot \color{blue}{\sin y} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{y} \cdot \cosh x\right) \cdot \sin \color{blue}{y} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{y} \cdot \cosh x\right), \color{blue}{\sin y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cosh x \cdot \frac{1}{y}\right), \sin \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cosh x}{y}\right), \sin \color{blue}{y}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\cosh x, y\right), \sin \color{blue}{y}\right) \]

      9. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), \sin y\right) \]

      10. sin-lowering-sin.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}, y\right), \mathsf{sin.f64}\left(y\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      4. *-lowering-*.f64 84.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

   7. Simplified 84.8%

      \[\leadsto \frac{\color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)}}{y} \cdot \sin y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), \color{blue}{y}\right) \]
   9. Step-by-step derivation

   10. Simplified 63.3%

       \[\leadsto \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y} \cdot \color{blue}{y} \]

   if 3.9499999999999998e90 < y < 1.15e203

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]
   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 81.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{4} \cdot \sin y}{y}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \left({x}^{4} \cdot \sin y\right)}{\color{blue}{y}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{4}\right) \cdot \sin y}{y} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{\left(2 \cdot 2\right)}\right) \cdot \sin y}{y} \]

      4. pow-sqr N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \sin y}{y} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \sin y}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \sin y}{y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right) \cdot \sin y\right), \color{blue}{y}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right), \sin y\right), y\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right), \sin y\right), y\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \sin y\right), y\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \sin y\right), y\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right), \sin y\right), y\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right), \sin y\right), y\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right), \sin y\right), y\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right), \sin y\right), y\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \sin y\right), y\right) \]

      17. sin-lowering-sin.f64 40.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right), \mathsf{sin.f64}\left(y\right)\right), y\right) \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \cdot \sin y}{y}} \]

   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

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \frac{-1}{144} \cdot \left({y}^{2} \cdot \color{blue}{{x}^{4}}\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{1}{24} \cdot {x}^{4} + \left(\frac{-1}{144} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}} \]

       4. distribute-rgt-out N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       13. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \frac{-1}{144} \cdot {y}^{2}\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{144} \cdot {y}^{2}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right)\right) \]

       21. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right)\right) \]

       22. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right)\right) \]

       23. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right)\right) \]

       24. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

       25. *-lowering-*.f64 34.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right)\right) \]

   11. Simplified 34.8%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{-1}{144} \cdot \left({x}^{4} \cdot {y}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot {y}^{2}\right) \cdot \color{blue}{\frac{-1}{144}} \]

       2. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left({y}^{2} \cdot \frac{-1}{144}\right)} \]

       3. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{-1}{144} \cdot \color{blue}{{y}^{2}}\right) \]

       4. metadata-eval N/A

          \[\leadsto {x}^{4} \cdot \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot {\color{blue}{y}}^{2}\right) \]

       5. associate-*r* N/A

          \[\leadsto {x}^{4} \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)}\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       8. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right), \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right), \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{24} \cdot \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right) \]

       14. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\left(\frac{1}{24} \cdot \frac{-1}{6}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{-1}{144} \cdot {\color{blue}{y}}^{2}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left({y}^{2} \cdot \color{blue}{\frac{-1}{144}}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\left(y \cdot y\right) \cdot \frac{-1}{144}\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \frac{-1}{144}\right)}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \left(y \cdot \left(\frac{-1}{144} \cdot \color{blue}{y}\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{-1}{144} \cdot y\right)}\right)\right) \]

       21. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

       22. *-lowering-*.f64 34.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{-1}{144}}\right)\right)\right) \]

   14. Simplified 34.8%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot -0.006944444444444444\right)\right)} \]

   if 1.15e203 < y

   1. Initial program 99.6%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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 85.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 85.7%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 16.8%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 19.1%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 19.1%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{+90}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;\left(y \cdot \left(y \cdot -0.006944444444444444\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.5e+20)
   (* y (/ (+ 1.0 (* (* x x) 0.5)) y))
   (* x (* 0.041666666666666664 (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.5e+20) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.5d+20) then
        tmp = y * ((1.0d0 + ((x * x) * 0.5d0)) / y)
    else
        tmp = x * (0.041666666666666664d0 * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.5e+20) {
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.5e+20:
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y)
	else:
		tmp = x * (0.041666666666666664 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.5e+20)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(Float64(x * x) * 0.5)) / y));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.5e+20)
		tmp = y * ((1.0 + ((x * x) * 0.5)) / y);
	else
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.5e+20], N[(y * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e20

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   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. associate-/r/ N/A

         \[\leadsto \cosh x \cdot \left(\frac{1}{y} \cdot \color{blue}{\sin y}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\cosh x \cdot \frac{1}{y}\right) \cdot \color{blue}{\sin y} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{y} \cdot \cosh x\right) \cdot \sin \color{blue}{y} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{y} \cdot \cosh x\right), \color{blue}{\sin y}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cosh x \cdot \frac{1}{y}\right), \sin \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cosh x}{y}\right), \sin \color{blue}{y}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\cosh x, y\right), \sin \color{blue}{y}\right) \]

      9. cosh-lowering-cosh.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), \sin y\right) \]

      10. sin-lowering-sin.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cosh.f64}\left(x\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}, y\right), \mathsf{sin.f64}\left(y\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

      4. *-lowering-*.f64 88.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), \mathsf{sin.f64}\left(y\right)\right) \]

   7. Simplified 88.5%

      \[\leadsto \frac{\color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)}}{y} \cdot \sin y \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), y\right), \color{blue}{y}\right) \]
   9. Step-by-step derivation

   10. Simplified 55.3%

       \[\leadsto \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y} \cdot \color{blue}{y} \]

   if 1.5e20 < 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 75.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 56.9%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 56.9%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 56.9%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;y \cdot \frac{1 + \left(x \cdot x\right) \cdot 0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 82:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 82.0)
   (+ 1.0 (* (* x x) 0.5))
   (* x (* 0.041666666666666664 (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 82.0) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 82.0d0) then
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    else
        tmp = x * (0.041666666666666664d0 * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 82.0) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 82.0:
		tmp = 1.0 + ((x * x) * 0.5)
	else:
		tmp = x * (0.041666666666666664 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 82.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(x * Float64(0.041666666666666664 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 82.0)
		tmp = 1.0 + ((x * x) * 0.5);
	else
		tmp = x * (0.041666666666666664 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 82.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 82

   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 63.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   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 53.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 53.3%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   if 82 < 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 68.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right)\right) \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)} \cdot \frac{\sin y}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right)\right)\right), \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 51.6%

      \[\leadsto \left(1 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right) \cdot \color{blue}{1} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   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. unpow2 N/A

          \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

       6. *-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \]

       10. unpow3 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{\color{blue}{3}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 51.6%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 51.6%

       \[\leadsto \color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 82:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 40.0) 1.0 (* (* x x) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 40.0) {
		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 <= 40.0d0) 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 <= 40.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 40.0:
		tmp = 1.0
	else:
		tmp = (x * x) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 40.0)
		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 <= 40.0)
		tmp = 1.0;
	else
		tmp = (x * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 40.0], 1.0, N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 40

   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 63.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 41.6%

      \[\leadsto \color{blue}{1} \]

   if 40 < 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

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot {x}^{2}} \]
   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 33.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   10. 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 33.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   11. Simplified 33.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 44.7% 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 68.6%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

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 48.0%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

8. Simplified 48.0%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

9. Final simplification 48.0%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot 0.5 \]
10. Add Preprocessing

Alternative 21: 26.3% 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 68.6%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 31.0%

   \[\leadsto \color{blue}{1} \]
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 2024161 
(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)))