Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 18

Speedup: 1.0×

• 50.9% 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: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.02:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.02) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.02) {
		tmp = cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 1.02d0) then
        tmp = cos(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 1.02) {
		tmp = Math.cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.02:
		tmp = math.cos(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.02)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.02)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 1.02], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.02

   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 99.3%

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

   5. Simplified 99.3%

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

   if 1.02 < (/.f64 (sinh.f64 y) 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 75.0%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 75.0%

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

   7. Applied egg-rr 75.0%

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

Alternative 3: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;y \leq 0.016:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ 1.0 (* y (* y 0.16666666666666666))))))
   (if (<= y 0.016)
     t_0
     (if (<= y 2e+152) (* (/ (sinh y) y) (+ 1.0 (* (* x x) -0.5))) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + (y * (y * 0.16666666666666666)));
	double tmp;
	if (y <= 0.016) {
		tmp = t_0;
	} else if (y <= 2e+152) {
		tmp = (sinh(y) / y) * (1.0 + ((x * x) * -0.5));
	} 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 <= 0.016d0) then
        tmp = t_0
    else if (y <= 2d+152) then
        tmp = (sinh(y) / y) * (1.0d0 + ((x * x) * (-0.5d0)))
    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 <= 0.016) {
		tmp = t_0;
	} else if (y <= 2e+152) {
		tmp = (Math.sinh(y) / y) * (1.0 + ((x * x) * -0.5));
	} 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 <= 0.016:
		tmp = t_0
	elif y <= 2e+152:
		tmp = (math.sinh(y) / y) * (1.0 + ((x * x) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))))
	tmp = 0.0
	if (y <= 0.016)
		tmp = t_0;
	elseif (y <= 2e+152)
		tmp = Float64(Float64(sinh(y) / y) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	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 <= 0.016)
		tmp = t_0;
	elseif (y <= 2e+152)
		tmp = (sinh(y) / y) * (1.0 + ((x * x) * -0.5));
	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[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.016], t$95$0, If[LessEqual[y, 2e+152], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.016 or 2.0000000000000001e152 < 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. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      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. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 85.4%

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

   5. Simplified 85.4%

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

   if 0.016 < y < 2.0000000000000001e152

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.9%

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

   5. Simplified 76.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.016:\\ \;\;\;\;\cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;y \leq 0.135:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ 1.0 (* y (* y 0.16666666666666666))))))
   (if (<= y 0.135) t_0 (if (<= y 3.3e+154) (* (sinh y) (/ 1.0 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 <= 0.135) {
		tmp = t_0;
	} else if (y <= 3.3e+154) {
		tmp = sinh(y) * (1.0 / 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 <= 0.135d0) then
        tmp = t_0
    else if (y <= 3.3d+154) then
        tmp = sinh(y) * (1.0d0 / 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 <= 0.135) {
		tmp = t_0;
	} else if (y <= 3.3e+154) {
		tmp = Math.sinh(y) * (1.0 / 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 <= 0.135:
		tmp = t_0
	elif y <= 3.3e+154:
		tmp = math.sinh(y) * (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666))))
	tmp = 0.0
	if (y <= 0.135)
		tmp = t_0;
	elseif (y <= 3.3e+154)
		tmp = Float64(sinh(y) * Float64(1.0 / 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 <= 0.135)
		tmp = t_0;
	elseif (y <= 3.3e+154)
		tmp = sinh(y) * (1.0 / 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[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.135], t$95$0, If[LessEqual[y, 3.3e+154], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.13500000000000001 or 3.3e154 < 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. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      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. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 85.4%

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

   5. Simplified 85.4%

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

   if 0.13500000000000001 < y < 3.3e154

   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 73.1%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \sinh \color{blue}{y}\right) \]

      5. sinh-lowering-sinh.f64 73.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{sinh.f64}\left(y\right)\right) \]

   7. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \sinh y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.135:\\ \;\;\;\;\cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00048:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00048) (cos x) (* (sinh y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00048) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) * (1.0 / 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 <= 0.00048d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00048) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00048:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00048)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00048)
		tmp = cos(x);
	else
		tmp = sinh(y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00048], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000012e-4

   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 68.3%

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

   5. Simplified 68.3%

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

   if 4.80000000000000012e-4 < 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 72.7%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \sinh \color{blue}{y}\right) \]

      5. sinh-lowering-sinh.f64 72.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{sinh.f64}\left(y\right)\right) \]

   7. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \sinh y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00048:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot -0.5\\ \mathbf{if}\;y \leq 0.00038:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{y \cdot \left(1 + \frac{\left(y \cdot y\right) \cdot \left(0.027777777777777776 - t\_0 \cdot t\_0\right)}{0.16666666666666666 - t\_0}\right)}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + t\_1\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))
        (t_1 (* (* x x) -0.5)))
   (if (<= y 0.00038)
     (cos x)
     (if (<= y 2e+75)
       (/
        (*
         y
         (+
          1.0
          (/
           (* (* y y) (- 0.027777777777777776 (* t_0 t_0)))
           (- 0.16666666666666666 t_0))))
        y)
       (if (<= y 2e+260)
         (+
          (+
           1.0
           (*
            y
            (*
             y
             (*
              (+ 1.0 t_1)
              (+
               0.16666666666666666
               (*
                y
                (*
                 y
                 (+
                  0.008333333333333333
                  (* (* y y) 0.0001984126984126984)))))))))
          t_1)
         (+ 1.0 (* 0.16666666666666666 (* y y))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	double t_1 = (x * x) * -0.5;
	double tmp;
	if (y <= 0.00038) {
		tmp = cos(x);
	} else if (y <= 2e+75) {
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	} else if (y <= 2e+260) {
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * y) * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0)))
    t_1 = (x * x) * (-0.5d0)
    if (y <= 0.00038d0) then
        tmp = cos(x)
    else if (y <= 2d+75) then
        tmp = (y * (1.0d0 + (((y * y) * (0.027777777777777776d0 - (t_0 * t_0))) / (0.16666666666666666d0 - t_0)))) / y
    else if (y <= 2d+260) then
        tmp = (1.0d0 + (y * (y * ((1.0d0 + t_1) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))) + t_1
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	double t_1 = (x * x) * -0.5;
	double tmp;
	if (y <= 0.00038) {
		tmp = Math.cos(x);
	} else if (y <= 2e+75) {
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	} else if (y <= 2e+260) {
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))
	t_1 = (x * x) * -0.5
	tmp = 0
	if y <= 0.00038:
		tmp = math.cos(x)
	elif y <= 2e+75:
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y
	elif y <= 2e+260:
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1
	else:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))
	t_1 = Float64(Float64(x * x) * -0.5)
	tmp = 0.0
	if (y <= 0.00038)
		tmp = cos(x);
	elseif (y <= 2e+75)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(Float64(y * y) * Float64(0.027777777777777776 - Float64(t_0 * t_0))) / Float64(0.16666666666666666 - t_0)))) / y);
	elseif (y <= 2e+260)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(Float64(1.0 + t_1) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))) + t_1);
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	t_1 = (x * x) * -0.5;
	tmp = 0.0;
	if (y <= 0.00038)
		tmp = cos(x);
	elseif (y <= 2e+75)
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	elseif (y <= 2e+260)
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	else
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[y, 0.00038], N[Cos[x], $MachinePrecision], If[LessEqual[y, 2e+75], N[(N[(y * N[(1.0 + N[(N[(N[(y * y), $MachinePrecision] * N[(0.027777777777777776 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2e+260], N[(N[(1.0 + N[(y * N[(y * N[(N[(1.0 + t$95$1), $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]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.8000000000000002e-4

   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 68.3%

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

   5. Simplified 68.3%

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

   if 3.8000000000000002e-4 < y < 1.99999999999999985e75

   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. 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. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 20.7%

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

   8. Simplified 20.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 67.3%

       \[\leadsto 1 \cdot \frac{y \cdot \left(1 + \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(y \cdot y\right)}{0.16666666666666666 - \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)}}\right)}{y} \]

   if 1.99999999999999985e75 < y < 2.00000000000000013e260

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 88.9%

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

   if 2.00000000000000013e260 < 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 85.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. 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 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00038:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{y \cdot \left(1 + \frac{\left(y \cdot y\right) \cdot \left(0.027777777777777776 - \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}{0.16666666666666666 - \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)}\right)}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot -0.5\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{y \cdot \left(1 + \frac{\left(y \cdot y\right) \cdot \left(0.027777777777777776 - t\_0 \cdot t\_0\right)}{0.16666666666666666 - t\_0}\right)}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + t\_1\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+ 0.008333333333333333 (* y (* y 0.0001984126984126984)))))
        (t_1 (* (* x x) -0.5)))
   (if (<= y 6.6e+72)
     (/
      (*
       y
       (+
        1.0
        (/
         (* (* y y) (- 0.027777777777777776 (* t_0 t_0)))
         (- 0.16666666666666666 t_0))))
      y)
     (if (<= y 2e+260)
       (+
        (+
         1.0
         (*
          y
          (*
           y
           (*
            (+ 1.0 t_1)
            (+
             0.16666666666666666
             (*
              y
              (*
               y
               (+
                0.008333333333333333
                (* (* y y) 0.0001984126984126984)))))))))
        t_1)
       (+ 1.0 (* 0.16666666666666666 (* y y)))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	double t_1 = (x * x) * -0.5;
	double tmp;
	if (y <= 6.6e+72) {
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	} else if (y <= 2e+260) {
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * y) * (0.008333333333333333d0 + (y * (y * 0.0001984126984126984d0)))
    t_1 = (x * x) * (-0.5d0)
    if (y <= 6.6d+72) then
        tmp = (y * (1.0d0 + (((y * y) * (0.027777777777777776d0 - (t_0 * t_0))) / (0.16666666666666666d0 - t_0)))) / y
    else if (y <= 2d+260) then
        tmp = (1.0d0 + (y * (y * ((1.0d0 + t_1) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))) + t_1
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	double t_1 = (x * x) * -0.5;
	double tmp;
	if (y <= 6.6e+72) {
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	} else if (y <= 2e+260) {
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)))
	t_1 = (x * x) * -0.5
	tmp = 0
	if y <= 6.6e+72:
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y
	elif y <= 2e+260:
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1
	else:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(y * Float64(y * 0.0001984126984126984))))
	t_1 = Float64(Float64(x * x) * -0.5)
	tmp = 0.0
	if (y <= 6.6e+72)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(Float64(y * y) * Float64(0.027777777777777776 - Float64(t_0 * t_0))) / Float64(0.16666666666666666 - t_0)))) / y);
	elseif (y <= 2e+260)
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(Float64(1.0 + t_1) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))) + t_1);
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.008333333333333333 + (y * (y * 0.0001984126984126984)));
	t_1 = (x * x) * -0.5;
	tmp = 0.0;
	if (y <= 6.6e+72)
		tmp = (y * (1.0 + (((y * y) * (0.027777777777777776 - (t_0 * t_0))) / (0.16666666666666666 - t_0)))) / y;
	elseif (y <= 2e+260)
		tmp = (1.0 + (y * (y * ((1.0 + t_1) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_1;
	else
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(y * N[(y * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[y, 6.6e+72], N[(N[(y * N[(1.0 + N[(N[(N[(y * y), $MachinePrecision] * N[(0.027777777777777776 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2e+260], N[(N[(1.0 + N[(y * N[(y * N[(N[(1.0 + t$95$1), $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]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.6e72

   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 66.0%

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

   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. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.0%

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

   8. Simplified 61.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   10. Applied egg-rr 46.2%

       \[\leadsto 1 \cdot \frac{y \cdot \left(1 + \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(y \cdot y\right)}{0.16666666666666666 - \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)}}\right)}{y} \]

   if 6.6e72 < y < 2.00000000000000013e260

   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. *-commutative N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 88.9%

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

   if 2.00000000000000013e260 < 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 85.7%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. 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 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{y \cdot \left(1 + \frac{\left(y \cdot y\right) \cdot \left(0.027777777777777776 - \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)\right)\right)}{0.16666666666666666 - \left(y \cdot y\right) \cdot \left(0.008333333333333333 + y \cdot \left(y \cdot 0.0001984126984126984\right)\right)}\right)}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot -0.5\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 - y \cdot \left(y \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + t\_0\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) -0.5)))
   (if (<= x 2.8e+289)
     (/
      (*
       y
       (+
        1.0
        (*
         (* y y)
         (+
          0.16666666666666666
          (*
           y
           (*
            y
            (- 0.008333333333333333 (* y (* y -0.0001984126984126984)))))))))
      y)
     (+
      (+
       1.0
       (*
        y
        (*
         y
         (*
          (+ 1.0 t_0)
          (+
           0.16666666666666666
           (*
            y
            (*
             y
             (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * -0.5;
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	} else {
		tmp = (1.0 + (y * (y * ((1.0 + t_0) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (-0.5d0)
    if (x <= 2.8d+289) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 - (y * (y * (-0.0001984126984126984d0)))))))))) / y
    else
        tmp = (1.0d0 + (y * (y * ((1.0d0 + t_0) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) * -0.5;
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	} else {
		tmp = (1.0 + (y * (y * ((1.0 + t_0) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * -0.5
	tmp = 0
	if x <= 2.8e+289:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y
	else:
		tmp = (1.0 + (y * (y * ((1.0 + t_0) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * -0.5)
	tmp = 0.0
	if (x <= 2.8e+289)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 - Float64(y * Float64(y * -0.0001984126984126984))))))))) / y);
	else
		tmp = Float64(Float64(1.0 + Float64(y * Float64(y * Float64(Float64(1.0 + t_0) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * -0.5;
	tmp = 0.0;
	if (x <= 2.8e+289)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	else
		tmp = (1.0 + (y * (y * ((1.0 + t_0) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[x, 2.8e+289], N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 - N[(y * N[(y * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 + N[(y * N[(y * N[(N[(1.0 + t$95$0), $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]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999991e289

   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 67.9%

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

   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. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 63.7%

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

   8. Simplified 63.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      8. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      15. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{5040}\right)\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(\mathsf{neg}\left(\frac{1}{5040}\right)\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{5040}\right)\right), \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

   10. Applied egg-rr 46.5%

       \[\leadsto 1 \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \color{blue}{\frac{\left(6.944444444444444 \cdot 10^{-5} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot 3.936759889140842 \cdot 10^{-8}\right) \cdot y}{0.008333333333333333 + -0.0001984126984126984 \cdot \left(y \cdot y\right)}}\right)\right)}{y} \]
   11. Step-by-step derivation

       1. *-lft-identity N/A

          \[\leadsto \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + y \cdot \frac{\left(\frac{1}{14400} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{25401600}\right) \cdot y}{\frac{1}{120} + \frac{-1}{5040} \cdot \left(y \cdot y\right)}\right)\right)}{\color{blue}{y}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + y \cdot \frac{\left(\frac{1}{14400} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{25401600}\right) \cdot y}{\frac{1}{120} + \frac{-1}{5040} \cdot \left(y \cdot y\right)}\right)\right)\right), \color{blue}{y}\right) \]

   12. Applied egg-rr 63.7%

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

   if 2.79999999999999991e289 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 41.4%

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

   5. Simplified 41.4%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.5 + \left(1 + y \cdot \left(y \cdot \left(\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 - y \cdot \left(y \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y \cdot \left(y \cdot \left(\left(1 + \left(x \cdot x\right) \cdot -0.5\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)\right)\right) + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 - y \cdot \left(y \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e+289)
   (/
    (*
     y
     (+
      1.0
      (*
       (* y y)
       (+
        0.16666666666666666
        (*
         y
         (* y (- 0.008333333333333333 (* y (* y -0.0001984126984126984)))))))))
    y)
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.8d+289) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 - (y * (y * (-0.0001984126984126984d0)))))))))) / y
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e+289:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e+289)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(0.008333333333333333 - Float64(y * Float64(y * -0.0001984126984126984))))))))) / y);
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e+289)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 - (y * (y * -0.0001984126984126984))))))))) / y;
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e+289], N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(0.008333333333333333 - N[(y * N[(y * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999991e289

   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 67.9%

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

   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. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 63.7%

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

   8. Simplified 63.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      8. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} - \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      15. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \left(\frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{5040}\right)\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\left(\mathsf{neg}\left(\frac{1}{5040}\right)\right) \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{5040}\right)\right), \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{14400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, y\right)\right), \frac{1}{25401600}\right)\right), y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), y\right)\right) \]

   10. Applied egg-rr 46.5%

       \[\leadsto 1 \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \color{blue}{\frac{\left(6.944444444444444 \cdot 10^{-5} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot 3.936759889140842 \cdot 10^{-8}\right) \cdot y}{0.008333333333333333 + -0.0001984126984126984 \cdot \left(y \cdot y\right)}}\right)\right)}{y} \]
   11. Step-by-step derivation

       1. *-lft-identity N/A

          \[\leadsto \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + y \cdot \frac{\left(\frac{1}{14400} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{25401600}\right) \cdot y}{\frac{1}{120} + \frac{-1}{5040} \cdot \left(y \cdot y\right)}\right)\right)}{\color{blue}{y}} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{6} + y \cdot \frac{\left(\frac{1}{14400} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{25401600}\right) \cdot y}{\frac{1}{120} + \frac{-1}{5040} \cdot \left(y \cdot y\right)}\right)\right)\right), \color{blue}{y}\right) \]

   12. Applied egg-rr 63.7%

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

   if 2.79999999999999991e289 < 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 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.4%

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

   14. Simplified 41.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 - y \cdot \left(y \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e+289)
   (+
    1.0
    (*
     (* y y)
     (+
      0.16666666666666666
      (* y (* y (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.8d+289) then
        tmp = 1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e+289:
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e+289)
		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(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e+289)
		tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))));
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e+289], 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[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999991e289

   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 67.9%

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

   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. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 63.0%

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

   8. Simplified 63.0%

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

   if 2.79999999999999991e289 < 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 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.4%

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

   14. Simplified 41.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e+289)
   (/
    (*
     y
     (+
      1.0
      (* (* y y) (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))))
    y)
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.8d+289) then
        tmp = (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0)))))) / y
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e+289:
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))) / y
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e+289)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333)))))) / y);
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e+289)
		tmp = (y * (1.0 + ((y * y) * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))) / y;
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e+289], N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999991e289

   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 67.9%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 67.9%

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

   7. Applied egg-rr 67.9%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 61.3%

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

   10. Simplified 61.3%

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

   if 2.79999999999999991e289 < 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 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.4%

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

   14. Simplified 41.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+94}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2e+94)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (if (<= x 2.8e+289)
     (* x (* x (* (* x x) 0.041666666666666664)))
     (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+94) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else if (x <= 2.8e+289) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2d+94) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else if (x <= 2.8d+289) then
        tmp = x * (x * ((x * x) * 0.041666666666666664d0))
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2e+94) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else if (x <= 2.8e+289) {
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e+94:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	elif x <= 2.8e+289:
		tmp = x * (x * ((x * x) * 0.041666666666666664))
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+94)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	elseif (x <= 2.8e+289)
		tmp = Float64(x * Float64(x * Float64(Float64(x * x) * 0.041666666666666664)));
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2e+94)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	elseif (x <= 2.8e+289)
		tmp = x * (x * ((x * x) * 0.041666666666666664));
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+94], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+289], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2e94

   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 74.4%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. 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%

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

   10. Simplified 56.4%

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

   if 2e94 < x < 2.79999999999999991e289

   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 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 32.7%

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

   8. Simplified 32.7%

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

   9. Taylor expanded in x around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

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

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

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

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

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

       19. unpow2 N/A

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

       20. *-lowering-*.f64 32.7%

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

   11. Simplified 32.7%

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

   if 2.79999999999999991e289 < 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 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.4%

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

   14. Simplified 41.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+94}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e+289)
   (+ 1.0 (* y (* y (+ 0.16666666666666666 (* y (* y 0.008333333333333333))))))
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.8d+289) then
        tmp = 1.0d0 + (y * (y * (0.16666666666666666d0 + (y * (y * 0.008333333333333333d0)))))
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e+289) {
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e+289:
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333)))))
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e+289)
		tmp = Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(y * Float64(y * 0.008333333333333333))))));
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e+289)
		tmp = 1.0 + (y * (y * (0.16666666666666666 + (y * (y * 0.008333333333333333)))));
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e+289], N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(y * N[(y * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999991e289

   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 67.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \sinh \color{blue}{y}\right) \]

      5. sinh-lowering-sinh.f64 67.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{sinh.f64}\left(y\right)\right) \]

   7. Applied egg-rr 67.9%

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

   8. Taylor expanded in y around 0

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

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

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

      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. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 60.6%

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

   10. Simplified 60.6%

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

   if 2.79999999999999991e289 < 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 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 0.9%

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

   11. Simplified 0.9%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.4%

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

   14. Simplified 41.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+289}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e+237)
   (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) y)
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+237) {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.45d+237) then
        tmp = (y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / y
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+237) {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e+237:
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e+237)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / y);
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e+237)
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e+237], N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000005e237

   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 69.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 59.2%

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

   10. Simplified 59.2%

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

   if 1.45000000000000005e237 < 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 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 24.2%

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

   11. Simplified 24.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.8%

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

   14. Simplified 41.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3400000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+136}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3400000000.0)
   1.0
   (if (<= y 1.4e+136) (* (* x x) -0.5) (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3400000000.0) {
		tmp = 1.0;
	} else if (y <= 1.4e+136) {
		tmp = (x * x) * -0.5;
	} else {
		tmp = 0.16666666666666666 * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3400000000.0d0) then
        tmp = 1.0d0
    else if (y <= 1.4d+136) then
        tmp = (x * x) * (-0.5d0)
    else
        tmp = 0.16666666666666666d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3400000000.0) {
		tmp = 1.0;
	} else if (y <= 1.4e+136) {
		tmp = (x * x) * -0.5;
	} else {
		tmp = 0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3400000000.0:
		tmp = 1.0
	elif y <= 1.4e+136:
		tmp = (x * x) * -0.5
	else:
		tmp = 0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3400000000.0)
		tmp = 1.0;
	elseif (y <= 1.4e+136)
		tmp = Float64(Float64(x * x) * -0.5);
	else
		tmp = Float64(0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3400000000.0)
		tmp = 1.0;
	elseif (y <= 1.4e+136)
		tmp = (x * x) * -0.5;
	else
		tmp = 0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3400000000.0], 1.0, If[LessEqual[y, 1.4e+136], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.4e9

   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 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 40.9%

      \[\leadsto \color{blue}{1} \]

   if 3.4e9 < y < 1.4000000000000001e136

   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 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 12.7%

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

   11. Simplified 12.7%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 21.1%

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

   14. Simplified 21.1%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]

   if 1.4000000000000001e136 < 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 72.4%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 72.4%

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

   7. Applied egg-rr 72.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. 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 72.4%

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

   10. Simplified 72.4%

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

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]
   12. 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 72.4%

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

   13. Simplified 72.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3400000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+136}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e+237)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+237) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.45d+237) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+237) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e+237:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e+237)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e+237)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e+237], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45000000000000005e237

   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 69.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. 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 51.9%

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

   10. Simplified 51.9%

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

   if 1.45000000000000005e237 < 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 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. cube-mult N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. sub-neg N/A

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

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

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

       16. associate-*r/ N/A

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

       17. metadata-eval N/A

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

       18. distribute-neg-frac N/A

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

       19. metadata-eval N/A

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

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

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

       21. unpow2 N/A

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

       22. *-lowering-*.f64 24.2%

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

   11. Simplified 24.2%

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

   12. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 41.8%

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

   14. Simplified 41.8%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.1% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10500000000.0) 1.0 (* (* x x) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10500000000.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 10500000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 10500000000.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 10500000000.0:
		tmp = 1.0
	else:
		tmp = (x * x) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10500000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 10500000000.0)
		tmp = 1.0;
	else
		tmp = (x * x) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10500000000.0], 1.0, N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e10

   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 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 40.9%

      \[\leadsto \color{blue}{1} \]

   if 1.05e10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. cos-lowering-cos.f64 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 13.2%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 13.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{2} \cdot {x}^{2}\right), \left(\color{blue}{\frac{1}{24}} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{24}} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{24} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

       16. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right) \]

       18. distribute-neg-frac N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

       19. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\frac{-1}{2}}{{\color{blue}{x}}^{2}}\right)\right)\right) \]

       20. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       22. *-lowering-*.f64 11.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 11.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.041666666666666664 + \frac{-0.5}{x \cdot x}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   14. Simplified 18.1%

       \[\leadsto \color{blue}{-0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.5% 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 54.3%

      \[\leadsto \mathsf{cos.f64}\left(x\right) \]

5. Simplified 54.3%

   \[\leadsto \color{blue}{\cos x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 32.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))