Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• 50.2% 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: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.062 \lor \neg \left(y \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 0.062) (not (<= y 1.4e+154)))
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 0.062) || !(y <= 1.4e+154)) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 0.062d0) .or. (.not. (y <= 1.4d+154))) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 0.062) || !(y <= 1.4e+154)) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 0.062) or not (y <= 1.4e+154):
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 0.062) || !(y <= 1.4e+154))
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 0.062) || ~((y <= 1.4e+154)))
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 0.062], N[Not[LessEqual[y, 1.4e+154]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.062 or 1.4e154 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.5%

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

      1. unpow2 85.5%

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

   5. Applied egg-rr 85.5%

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

   if 0.062 < y < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. expm1-log1p-u 84.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      3. expm1-undefine 84.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]

   5. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 84.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} + \left(-1\right)} \]

      2. metadata-eval 84.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} + \color{blue}{-1} \]

      3. +-commutative 84.6%

         \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)}} \]

      4. log1p-undefine 84.6%

         \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{\sinh y}{y}\right)}} \]

      5. rem-exp-log 84.8%

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

      6. associate-+r+ 84.8%

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

      7. metadata-eval 84.8%

         \[\leadsto \color{blue}{0} + \frac{\sinh y}{y} \]

      8. +-lft-identity 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.062 \lor \neg \left(y \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0053:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.0053) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0053) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0053d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0053) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0053:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0053)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0053)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0053], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00530000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.1%

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

   if 0.00530000000000000002 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.2%

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

      1. *-un-lft-identity 85.2%

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

      2. expm1-log1p-u 85.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      3. expm1-undefine 85.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   6. Step-by-step derivation

      1. sub-neg 85.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} + \left(-1\right)} \]

      2. metadata-eval 85.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} + \color{blue}{-1} \]

      3. +-commutative 85.1%

         \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)}} \]

      4. log1p-undefine 85.1%

         \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{\sinh y}{y}\right)}} \]

      5. rem-exp-log 85.2%

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

      6. associate-+r+ 85.2%

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

      7. metadata-eval 85.2%

         \[\leadsto \color{blue}{0} + \frac{\sinh y}{y} \]

      8. +-lft-identity 85.2%

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

   7. Simplified 85.2%

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

Alternative 4: 61.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 245:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 245.0)
     (cos x)
     (if (<= y 4e+163) (* t_0 (+ 1.0 (* (* x x) -0.5))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 245.0) {
		tmp = cos(x);
	} else if (y <= 4e+163) {
		tmp = t_0 * (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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 245.0d0) then
        tmp = cos(x)
    else if (y <= 4d+163) then
        tmp = t_0 * (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 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 245.0) {
		tmp = Math.cos(x);
	} else if (y <= 4e+163) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 245.0:
		tmp = math.cos(x)
	elif y <= 4e+163:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 245.0)
		tmp = cos(x);
	elseif (y <= 4e+163)
		tmp = Float64(t_0 * 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 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 245.0)
		tmp = cos(x);
	elseif (y <= 4e+163)
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	else
		tmp = t_0;
	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, 245.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 4e+163], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 245

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.1%

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

   if 245 < y < 3.9999999999999998e163

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 11.2%

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

      1. unpow2 11.2%

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

   5. Applied egg-rr 11.2%

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

   6. Taylor expanded in x around 0 20.6%

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

      1. *-commutative 20.6%

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

   8. Simplified 20.6%

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

      1. unpow2 20.6%

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

   10. Applied egg-rr 20.6%

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

   if 3.9999999999999998e163 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.5%

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

   4. Taylor expanded in y around 0 88.5%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 88.5%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 245:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+163}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+163}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (<= y 1.1e-8)
     1.0
     (if (<= y 6e+163) (* t_0 (+ 1.0 (* (* x x) -0.5))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.1e-8) {
		tmp = 1.0;
	} else if (y <= 6e+163) {
		tmp = t_0 * (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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if (y <= 1.1d-8) then
        tmp = 1.0d0
    else if (y <= 6d+163) then
        tmp = t_0 * (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 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if (y <= 1.1e-8) {
		tmp = 1.0;
	} else if (y <= 6e+163) {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if y <= 1.1e-8:
		tmp = 1.0
	elif y <= 6e+163:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (y <= 1.1e-8)
		tmp = 1.0;
	elseif (y <= 6e+163)
		tmp = Float64(t_0 * 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 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (y <= 1.1e-8)
		tmp = 1.0;
	elseif (y <= 6e+163)
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	else
		tmp = t_0;
	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, 1.1e-8], 1.0, If[LessEqual[y, 6e+163], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.0999999999999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 53.8%

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

   4. Taylor expanded in y around 0 30.2%

      \[\leadsto \color{blue}{1} \]

   if 1.0999999999999999e-8 < y < 6.00000000000000027e163

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 18.1%

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

      1. unpow2 18.1%

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

   5. Applied egg-rr 18.1%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. *-commutative 23.6%

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

   8. Simplified 23.6%

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

      1. unpow2 23.6%

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

   10. Applied egg-rr 23.6%

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

   if 6.00000000000000027e163 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.5%

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

   4. Taylor expanded in y around 0 88.5%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 88.5%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-8}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+163}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.9% 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 x around 0 61.2%

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

4. Taylor expanded in y around 0 42.5%

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

   1. unpow2 75.2%

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

6. Applied egg-rr 42.5%

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

Alternative 7: 28.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 61.2%

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

4. Taylor expanded in y around 0 23.7%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))