Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 14.0s

Alternatives: 20

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\\ \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot t\_0\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(y \cdot \left(1 + y \cdot \left(y \cdot t\_0\right)\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))
   (if (<= y 0.215)
     (* (cos x) (+ 1.0 (* (* y y) t_0)))
     (if (<= y 1.15e+62)
       (/ (sinh y) y)
       (/ (* (cos x) (* y (+ 1.0 (* y (* y t_0))))) y)))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333);
	double tmp;
	if (y <= 0.215) {
		tmp = cos(x) * (1.0 + ((y * y) * t_0));
	} else if (y <= 1.15e+62) {
		tmp = sinh(y) / y;
	} else {
		tmp = (cos(x) * (y * (1.0 + (y * (y * t_0))))) / y;
	}
	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 = 0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0)
    if (y <= 0.215d0) then
        tmp = cos(x) * (1.0d0 + ((y * y) * t_0))
    else if (y <= 1.15d+62) then
        tmp = sinh(y) / y
    else
        tmp = (cos(x) * (y * (1.0d0 + (y * (y * t_0))))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333);
	double tmp;
	if (y <= 0.215) {
		tmp = Math.cos(x) * (1.0 + ((y * y) * t_0));
	} else if (y <= 1.15e+62) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = (Math.cos(x) * (y * (1.0 + (y * (y * t_0))))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333)
	tmp = 0
	if y <= 0.215:
		tmp = math.cos(x) * (1.0 + ((y * y) * t_0))
	elif y <= 1.15e+62:
		tmp = math.sinh(y) / y
	else:
		tmp = (math.cos(x) * (y * (1.0 + (y * (y * t_0))))) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))
	tmp = 0.0
	if (y <= 0.215)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * t_0)));
	elseif (y <= 1.15e+62)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(Float64(cos(x) * Float64(y * Float64(1.0 + Float64(y * Float64(y * t_0))))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 + ((y * y) * 0.008333333333333333);
	tmp = 0.0;
	if (y <= 0.215)
		tmp = cos(x) * (1.0 + ((y * y) * t_0));
	elseif (y <= 1.15e+62)
		tmp = sinh(y) / y;
	else
		tmp = (cos(x) * (y * (1.0 + (y * (y * t_0))))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.215], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(y * N[(1.0 + N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.214999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 91.0%

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

   if 0.214999999999999997 < y < 1.14999999999999992e62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 87.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 87.5%

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

   7. Applied egg-rr 87.5%

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

   if 1.14999999999999992e62 < y

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \frac{\cos x \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)\right)}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{if}\;y \leq 0.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
   (if (<= y 0.4) t_0 (if (<= y 3.8e+77) (/ (sinh y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	double tmp;
	if (y <= 0.4) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(x) * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))
    if (y <= 0.4d0) then
        tmp = t_0
    else if (y <= 3.8d+77) then
        tmp = sinh(y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	double tmp;
	if (y <= 0.4) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))
	tmp = 0
	if y <= 0.4:
		tmp = t_0
	elif y <= 3.8e+77:
		tmp = math.sinh(y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))
	tmp = 0.0
	if (y <= 0.4)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = Float64(sinh(y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	tmp = 0.0;
	if (y <= 0.4)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = sinh(y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.4], t$95$0, If[LessEqual[y, 3.8e+77], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.40000000000000002 or 3.8000000000000001e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

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

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

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

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

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

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

   5. Simplified 93.0%

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

   if 0.40000000000000002 < y < 3.8000000000000001e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 84.6%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 84.6%

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

   7. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{if}\;y \leq 0.0135:\\ \;\;\;\;\cos x \cdot t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\cos x \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* y y) 0.16666666666666666))))
   (if (<= y 0.0135)
     (* (cos x) t_0)
     (if (<= y 1.3e+103) (/ (sinh y) y) (/ (* t_0 (* (cos x) y)) y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 0.0135) {
		tmp = cos(x) * t_0;
	} else if (y <= 1.3e+103) {
		tmp = sinh(y) / y;
	} else {
		tmp = (t_0 * (cos(x) * y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    if (y <= 0.0135d0) then
        tmp = cos(x) * t_0
    else if (y <= 1.3d+103) then
        tmp = sinh(y) / y
    else
        tmp = (t_0 * (cos(x) * y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	double tmp;
	if (y <= 0.0135) {
		tmp = Math.cos(x) * t_0;
	} else if (y <= 1.3e+103) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = (t_0 * (Math.cos(x) * y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((y * y) * 0.16666666666666666)
	tmp = 0
	if y <= 0.0135:
		tmp = math.cos(x) * t_0
	elif y <= 1.3e+103:
		tmp = math.sinh(y) / y
	else:
		tmp = (t_0 * (math.cos(x) * y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))
	tmp = 0.0
	if (y <= 0.0135)
		tmp = Float64(cos(x) * t_0);
	elseif (y <= 1.3e+103)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(Float64(t_0 * Float64(cos(x) * y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((y * y) * 0.16666666666666666);
	tmp = 0.0;
	if (y <= 0.0135)
		tmp = cos(x) * t_0;
	elseif (y <= 1.3e+103)
		tmp = sinh(y) / y;
	else
		tmp = (t_0 * (cos(x) * y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.0135], N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 1.3e+103], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0134999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 81.8%

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

   5. Simplified 81.8%

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

   if 0.0134999999999999998 < y < 1.3000000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 83.3%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 83.3%

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

   7. Applied egg-rr 83.3%

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

   if 1.3000000000000001e103 < y

   1. Initial program 100.0%

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

      5. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

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

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

   7. Simplified 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0135:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\cos x \cdot y\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq 0.03:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ 1.0 (* (* y y) 0.16666666666666666)))))
   (if (<= y 0.03) t_0 (if (<= y 1.1e+154) (/ (sinh y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	double tmp;
	if (y <= 0.03) {
		tmp = t_0;
	} else if (y <= 1.1e+154) {
		tmp = sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(x) * (1.0d0 + ((y * y) * 0.16666666666666666d0))
    if (y <= 0.03d0) then
        tmp = t_0
    else if (y <= 1.1d+154) then
        tmp = sinh(y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	double tmp;
	if (y <= 0.03) {
		tmp = t_0;
	} else if (y <= 1.1e+154) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (1.0 + ((y * y) * 0.16666666666666666))
	tmp = 0
	if y <= 0.03:
		tmp = t_0
	elif y <= 1.1e+154:
		tmp = math.sinh(y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))
	tmp = 0.0
	if (y <= 0.03)
		tmp = t_0;
	elseif (y <= 1.1e+154)
		tmp = Float64(sinh(y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	tmp = 0.0;
	if (y <= 0.03)
		tmp = t_0;
	elseif (y <= 1.1e+154)
		tmp = sinh(y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.03], t$95$0, If[LessEqual[y, 1.1e+154], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.029999999999999999 or 1.1000000000000001e154 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-lft-identity N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.2%

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

   5. Simplified 84.2%

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

   if 0.029999999999999999 < y < 1.1000000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 78.1%

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

   7. Applied egg-rr 78.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.03:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0076:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.0076) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0076) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0076d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0076) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0076:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0076)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0076)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0076], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00759999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 64.4%

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

   5. Simplified 64.4%

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

   if 0.00759999999999999998 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 74.2%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

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

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

      3. sinh-lowering-sinh.f64 74.2%

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

   7. Applied egg-rr 74.2%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\\ \mathbf{if}\;y \leq 5.8:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\frac{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}{1 - t\_0 \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+
           0.16666666666666666
           (*
            (* y y)
            (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
   (if (<= y 5.8)
     (cos x)
     (if (<= y 1.35e+106)
       (/ 1.0 (/ (+ 1.0 (* y (* y -0.16666666666666666))) (- 1.0 (* t_0 t_0))))
       (*
        (+
         1.0
         (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))
        (+ 1.0 (* x (* x -0.5))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 5.8) {
		tmp = cos(x);
	} else if (y <= 1.35e+106) {
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	} else {
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))
    if (y <= 5.8d0) then
        tmp = cos(x)
    else if (y <= 1.35d+106) then
        tmp = 1.0d0 / ((1.0d0 + (y * (y * (-0.16666666666666666d0)))) / (1.0d0 - (t_0 * t_0)))
    else
        tmp = (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))) * (1.0d0 + (x * (x * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 5.8) {
		tmp = Math.cos(x);
	} else if (y <= 1.35e+106) {
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	} else {
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))
	tmp = 0
	if y <= 5.8:
		tmp = math.cos(x)
	elif y <= 1.35e+106:
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)))
	else:
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))
	tmp = 0.0
	if (y <= 5.8)
		tmp = cos(x);
	elseif (y <= 1.35e+106)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) / Float64(1.0 - Float64(t_0 * t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))) * Float64(1.0 + Float64(x * Float64(x * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	tmp = 0.0;
	if (y <= 5.8)
		tmp = cos(x);
	elseif (y <= 1.35e+106)
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	else
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.8], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.35e+106], N[(1.0 / N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.79999999999999982

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 64.4%

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

   5. Simplified 64.4%

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

   if 5.79999999999999982 < y < 1.35000000000000003e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 85.0%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 57.1%

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

   8. Simplified 57.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 6.9%

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

   11. Taylor expanded in y around 0

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-lowering-*.f64 61.8%

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

   13. Simplified 61.8%

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

   if 1.35000000000000003e106 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 82.6%

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

   8. Simplified 82.6%

      \[\leadsto \left(1 + x \cdot \left(x \cdot -0.5\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\frac{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}{1 - \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\\ \mathbf{if}\;y \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\frac{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}{1 - t\_0 \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+
           0.16666666666666666
           (*
            (* y y)
            (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
   (if (<= y 7e+106)
     (/ 1.0 (/ (+ 1.0 (* y (* y -0.16666666666666666))) (- 1.0 (* t_0 t_0))))
     (*
      (+
       1.0
       (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))
      (+ 1.0 (* x (* x -0.5)))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 7e+106) {
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	} else {
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))
    if (y <= 7d+106) then
        tmp = 1.0d0 / ((1.0d0 + (y * (y * (-0.16666666666666666d0)))) / (1.0d0 - (t_0 * t_0)))
    else
        tmp = (1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))) * (1.0d0 + (x * (x * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	double tmp;
	if (y <= 7e+106) {
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	} else {
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))
	tmp = 0
	if y <= 7e+106:
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)))
	else:
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))
	tmp = 0.0
	if (y <= 7e+106)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))) / Float64(1.0 - Float64(t_0 * t_0))));
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))) * Float64(1.0 + Float64(x * Float64(x * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))));
	tmp = 0.0;
	if (y <= 7e+106)
		tmp = 1.0 / ((1.0 + (y * (y * -0.16666666666666666))) / (1.0 - (t_0 * t_0)));
	else
		tmp = (1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))) * (1.0 + (x * (x * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7e+106], N[(1.0 / N[(N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999962e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 63.6%

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

   8. Simplified 63.6%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 40.5%

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

   11. Taylor expanded in y around 0

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-lowering-*.f64 53.8%

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

   13. Simplified 53.8%

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

   if 6.99999999999999962e106 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 82.6%

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

   8. Simplified 82.6%

      \[\leadsto \left(1 + x \cdot \left(x \cdot -0.5\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{\frac{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}{1 - \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right) \cdot \left(1 + x \cdot \left(x \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\\ t_1 := \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot t\_0\right)\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{+258}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot t\_0\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{1 - t\_1 \cdot t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))
        (t_1 (* (* y y) (+ 0.16666666666666666 (* (* y y) t_0)))))
   (if (<= x 8.5e+258)
     (/ (* y (+ 1.0 (* (* y y) (+ 0.16666666666666666 (* y (* y t_0)))))) y)
     (/ 1.0 (/ 1.0 (- 1.0 (* t_1 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 0.008333333333333333 + ((y * y) * 0.0001984126984126984);
	double t_1 = (y * y) * (0.16666666666666666 + ((y * y) * t_0));
	double tmp;
	if (x <= 8.5e+258) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * t_0)))))) / y;
	} else {
		tmp = 1.0 / (1.0 / (1.0 - (t_1 * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)
    t_1 = (y * y) * (0.16666666666666666d0 + ((y * y) * t_0))
    if (x <= 8.5d+258) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * t_0)))))) / y
    else
        tmp = 1.0d0 / (1.0d0 / (1.0d0 - (t_1 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.008333333333333333 + ((y * y) * 0.0001984126984126984);
	double t_1 = (y * y) * (0.16666666666666666 + ((y * y) * t_0));
	double tmp;
	if (x <= 8.5e+258) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * t_0)))))) / y;
	} else {
		tmp = 1.0 / (1.0 / (1.0 - (t_1 * t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.008333333333333333 + ((y * y) * 0.0001984126984126984)
	t_1 = (y * y) * (0.16666666666666666 + ((y * y) * t_0))
	tmp = 0
	if x <= 8.5e+258:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * t_0)))))) / y
	else:
		tmp = 1.0 / (1.0 / (1.0 - (t_1 * t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))
	t_1 = Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * t_0)))
	tmp = 0.0
	if (x <= 8.5e+258)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * t_0)))))) / y);
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(1.0 - Float64(t_1 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.008333333333333333 + ((y * y) * 0.0001984126984126984);
	t_1 = (y * y) * (0.16666666666666666 + ((y * y) * t_0));
	tmp = 0.0;
	if (x <= 8.5e+258)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * t_0)))))) / y;
	else
		tmp = 1.0 / (1.0 / (1.0 - (t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 8.5e+258], N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(1.0 / N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999974e258

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.4%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 68.8%

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

   8. Simplified 68.8%

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

   if 8.49999999999999974e258 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 23.3%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 15.9%

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

   8. Simplified 15.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 8.2%

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

   11. Taylor expanded in y around 0

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

   13. Simplified 46.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+258}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (/
    (*
     y
     (+
      1.0
      (*
       (* y y)
       (+
        0.16666666666666666
        (*
         y
         (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))
    y)
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / y;
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))))) / y
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / y;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / y
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))) / y);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))))) / y;
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 68.6%

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

   8. Simplified 68.6%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (+
    1.0
    (*
     (* y y)
     (+
      0.16666666666666666
      (* y (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = 1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.1%

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

   8. Simplified 67.1%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.9% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (+
    1.0
    (*
     (* y y)
     (+ 0.16666666666666666 (* y (* y (* (* y y) 0.0001984126984126984))))))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = 1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * ((y * y) * 0.0001984126984126984d0)))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))))
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.1%

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

   8. Simplified 67.1%

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 67.1%

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 67.1%

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

   13. Simplified 67.1%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (+ 1.0 (* (* y y) (* y (* y (* (* y y) 0.0001984126984126984)))))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = 1.0d0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984d0))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))))
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(1.0 + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.1%

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

   8. Simplified 67.1%

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 67.1%

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

   10. Applied egg-rr 67.1%

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

   11. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 66.9%

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

   13. Simplified 66.9%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (+ 1.0 (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = 1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 63.2%

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

   8. Simplified 63.2%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.8e+261)
   (+ 1.0 (* y (* y (* (* y y) 0.008333333333333333))))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 5.8d+261) then
        tmp = 1.0d0 + (y * (y * ((y * y) * 0.008333333333333333d0)))
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.8e+261) {
		tmp = 1.0 + (y * (y * ((y * y) * 0.008333333333333333)));
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.8e+261:
		tmp = 1.0 + (y * (y * ((y * y) * 0.008333333333333333)))
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.8e+261)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.8e+261)
		tmp = 1.0 + (y * (y * ((y * y) * 0.008333333333333333)));
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.8e+261], N[(1.0 + N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 63.2%

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

   8. Simplified 63.2%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 63.1%

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

   11. Simplified 63.1%

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

   if 5.8e261 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.5%

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

   5. Simplified 43.5%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 42.4%

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

   8. Simplified 42.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+261}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0075:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0075)
   (+ 1.0 (* (* y y) 0.16666666666666666))
   (* y (* y (* (* y y) 0.008333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0075) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = y * (y * ((y * y) * 0.008333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0075d0) then
        tmp = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    else
        tmp = y * (y * ((y * y) * 0.008333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0075) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = y * (y * ((y * y) * 0.008333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0075:
		tmp = 1.0 + ((y * y) * 0.16666666666666666)
	else:
		tmp = y * (y * ((y * y) * 0.008333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0075)
		tmp = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = Float64(y * Float64(y * Float64(Float64(y * y) * 0.008333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0075)
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	else
		tmp = y * (y * ((y * y) * 0.008333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0075], N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0074999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 70.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 56.0%

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

   8. Simplified 56.0%

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

   if 0.0074999999999999997 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{120} \cdot {y}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 57.9%

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

   11. Simplified 57.9%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0075:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+163}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.8e+163)
   (+ 1.0 (* (* y y) 0.16666666666666666))
   (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.8e+163) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = 1.0 + (-0.5 * (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 <= 9.8d+163) then
        tmp = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9.8e+163) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9.8e+163:
		tmp = 1.0 + ((y * y) * 0.16666666666666666)
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.8e+163)
		tmp = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9.8e+163)
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.8e+163], N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.8e163

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 76.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 56.4%

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

   8. Simplified 56.4%

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

   if 9.8e163 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 43.4%

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

   5. Simplified 43.4%

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

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 37.4%

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

   8. Simplified 37.4%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{+163}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.5% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0075:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0075) 1.0 (* (* y y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0075) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0075d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0075) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0075:
		tmp = 1.0
	else:
		tmp = (y * y) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0075)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0075)
		tmp = 1.0;
	else
		tmp = (y * y) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0075], 1.0, N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0074999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 64.5%

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

   5. Simplified 64.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 41.2%

      \[\leadsto \color{blue}{1} \]

   if 0.0074999999999999997 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 73.2%

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

   6. Taylor expanded in y around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in y around 0

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 38.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   11. Simplified 38.2%

       \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right) \]

       3. *-lowering-*.f64 38.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   14. Simplified 38.2%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0075:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* y y) 0.16666666666666666)))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * 0.16666666666666666);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * 0.16666666666666666d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * 0.16666666666666666);
}

Python

def code(x, y):
	return 1.0 + ((y * y) * 0.16666666666666666)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * 0.16666666666666666);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), y\right)\right) \]
4. Step-by-step derivation

5. Simplified 70.9%

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   4. *-lowering-*.f64 51.3%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

8. Simplified 51.3%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

9. Final simplification 51.3%

   \[\leadsto 1 + \left(y \cdot y\right) \cdot 0.16666666666666666 \]
10. Add Preprocessing

Alternative 20: 28.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 48.6%

      \[\leadsto \mathsf{cos.f64}\left(x\right) \]

5. Simplified 48.6%

   \[\leadsto \color{blue}{\cos x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 31.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))