Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 8

Speedup: 1.0×

• 50.5% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]
5. Add Preprocessing

Alternative 2: 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 3: 78.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{+20} \lor \neg \left(y \leq 1.65 \cdot 10^{+153}\right):\\ \;\;\;\;\cos x \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (or (<= y 2.6e+20) (not (<= y 1.65e+153)))
     (* (cos x) t_0)
     (* t_0 (+ 1.0 (* (* x x) -0.5))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if ((y <= 2.6e+20) || !(y <= 1.65e+153)) {
		tmp = cos(x) * t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if ((y <= 2.6d+20) .or. (.not. (y <= 1.65d+153))) then
        tmp = cos(x) * t_0
    else
        tmp = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if ((y <= 2.6e+20) || !(y <= 1.65e+153)) {
		tmp = Math.cos(x) * t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if (y <= 2.6e+20) or not (y <= 1.65e+153):
		tmp = math.cos(x) * t_0
	else:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if ((y <= 2.6e+20) || !(y <= 1.65e+153))
		tmp = Float64(cos(x) * t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if ((y <= 2.6e+20) || ~((y <= 1.65e+153)))
		tmp = cos(x) * t_0;
	else
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, 2.6e+20], N[Not[LessEqual[y, 1.65e+153]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e20 or 1.64999999999999997e153 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 82.3%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 82.3%

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

   5. Applied egg-rr 82.3%

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

   if 2.6e20 < y < 1.64999999999999997e153

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 5.6%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 5.6%

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

   5. Applied egg-rr 5.6%

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

   6. Taylor expanded in x around 0 28.9%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 28.9%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 28.9%

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

   10. Applied egg-rr 28.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+20} \lor \neg \left(y \leq 1.65 \cdot 10^{+153}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.1e+20)
   (cos x)
   (* (+ 1.0 (* 0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.1e+20) {
		tmp = cos(x);
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.1d+20) then
        tmp = cos(x)
    else
        tmp = (1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.1e+20) {
		tmp = Math.cos(x);
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.1e+20:
		tmp = math.cos(x)
	else:
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.1e+20)
		tmp = cos(x);
	else
		tmp = Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.1e+20)
		tmp = cos(x);
	else
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.1e+20], N[Cos[x], $MachinePrecision], N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.3%

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

   if 4.1e20 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 53.7%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 53.7%

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

   5. Applied egg-rr 53.7%

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

   6. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Simplified 53.8%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 53.8%

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

   10. Applied egg-rr 53.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+20}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 2.9e+20) t_0 (* t_0 (+ 1.0 (* (* x x) -0.5))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 2.9e+20) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 2.9d+20) then
        tmp = t_0
    else
        tmp = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 2.9e+20) {
		tmp = t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 2.9e+20:
		tmp = t_0
	else:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 2.9e+20)
		tmp = t_0;
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 2.9e+20)
		tmp = t_0;
	else
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.9e+20], t$95$0, N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

   4. Taylor expanded in x around 0 50.9%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 79.9%

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

   6. Applied egg-rr 50.9%

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

   if 2.9e20 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 53.7%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. unpow2 53.7%

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

   5. Applied egg-rr 53.7%

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

   6. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Simplified 53.8%

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 53.8%

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

   10. Applied egg-rr 53.8%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+144}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e+144)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+144) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.45d+144) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = 1.0d0 + ((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+144) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e+144:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e+144)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e+144)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e+144], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999999e144

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

   4. Taylor expanded in x around 0 51.0%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 74.8%

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

   6. Applied egg-rr 51.0%

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

   if 1.44999999999999999e144 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 39.0%

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

   4. Taylor expanded in x around 0 37.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 37.7%

         \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   6. Simplified 37.7%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 37.7%

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

   8. Applied egg-rr 37.7%

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

Alternative 7: 47.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.16666666666666666d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 74.5%

   \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

4. Taylor expanded in x around 0 47.7%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
5. Step-by-step derivation

   1. unpow2 74.5%

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

6. Applied egg-rr 47.7%

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

Alternative 8: 28.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 74.5%

   \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

4. Taylor expanded in x around 0 47.7%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]

5. Taylor expanded in y around 0 29.2%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))