Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 9

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. Final simplification 100.0%

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

Alternative 2: 63.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1700:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1700.0)
   (cos x)
   (if (<= y 5.5e+81)
     (/ (+ y (* -0.5 (* y (* x x)))) y)
     (/ (+ y (* 0.16666666666666666 (pow y 3.0))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1700.0) {
		tmp = cos(x);
	} else if (y <= 5.5e+81) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = (y + (0.16666666666666666 * pow(y, 3.0))) / 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 <= 1700.0d0) then
        tmp = cos(x)
    else if (y <= 5.5d+81) then
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    else
        tmp = (y + (0.16666666666666666d0 * (y ** 3.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1700.0) {
		tmp = Math.cos(x);
	} else if (y <= 5.5e+81) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = (y + (0.16666666666666666 * Math.pow(y, 3.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1700.0:
		tmp = math.cos(x)
	elif y <= 5.5e+81:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	else:
		tmp = (y + (0.16666666666666666 * math.pow(y, 3.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1700.0)
		tmp = cos(x);
	elseif (y <= 5.5e+81)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	else
		tmp = Float64(Float64(y + Float64(0.16666666666666666 * (y ^ 3.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1700.0)
		tmp = cos(x);
	elseif (y <= 5.5e+81)
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	else
		tmp = (y + (0.16666666666666666 * (y ^ 3.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1700.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 5.5e+81], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1700

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 1700 < y < 5.5000000000000003e81

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 15.8%

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

      1. unpow2 15.8%

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

   8. Applied egg-rr 15.8%

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

   if 5.5000000000000003e81 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 88.9%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot \left({y}^{3} \cdot \cos x\right) + y \cdot \cos x}}{y} \]

   6. Taylor expanded in x around 0 65.7%

      \[\leadsto \color{blue}{\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1700:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 400:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 400.0)
   (cos x)
   (if (<= y 1.6e+155)
     (/ (+ y (* -0.5 (* y (* x x)))) y)
     (+ 1.0 (* 0.16666666666666666 (pow y 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 400.0) {
		tmp = cos(x);
	} else if (y <= 1.6e+155) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = 1.0 + (0.16666666666666666 * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 400.0d0) then
        tmp = cos(x)
    else if (y <= 1.6d+155) then
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 400.0) {
		tmp = Math.cos(x);
	} else if (y <= 1.6e+155) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = 1.0 + (0.16666666666666666 * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 400.0:
		tmp = math.cos(x)
	elif y <= 1.6e+155:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	else:
		tmp = 1.0 + (0.16666666666666666 * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 400.0)
		tmp = cos(x);
	elseif (y <= 1.6e+155)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 400.0)
		tmp = cos(x);
	elseif (y <= 1.6e+155)
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	else
		tmp = 1.0 + (0.16666666666666666 * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 400.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.6e+155], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 400

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 400 < y < 1.60000000000000006e155

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 20.9%

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

      1. unpow2 17.3%

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

   8. Applied egg-rr 20.9%

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

   if 1.60000000000000006e155 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot \left({y}^{3} \cdot \cos x\right) + y \cdot \cos x}}{y} \]

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}} \]

   7. Taylor expanded in y around 0 77.4%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 400:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 420:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 420.0)
   (cos x)
   (if (<= y 5.6e+81)
     (/ (+ y (* -0.5 (* y (* x x)))) y)
     (/ (* 0.16666666666666666 (pow y 3.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 420.0) {
		tmp = cos(x);
	} else if (y <= 5.6e+81) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = (0.16666666666666666 * pow(y, 3.0)) / 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 <= 420.0d0) then
        tmp = cos(x)
    else if (y <= 5.6d+81) then
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    else
        tmp = (0.16666666666666666d0 * (y ** 3.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 420.0) {
		tmp = Math.cos(x);
	} else if (y <= 5.6e+81) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = (0.16666666666666666 * Math.pow(y, 3.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 420.0:
		tmp = math.cos(x)
	elif y <= 5.6e+81:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	else:
		tmp = (0.16666666666666666 * math.pow(y, 3.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 420.0)
		tmp = cos(x);
	elseif (y <= 5.6e+81)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	else
		tmp = Float64(Float64(0.16666666666666666 * (y ^ 3.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 420.0)
		tmp = cos(x);
	elseif (y <= 5.6e+81)
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	else
		tmp = (0.16666666666666666 * (y ^ 3.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 420.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 5.6e+81], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 420

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 420 < y < 5.5999999999999999e81

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 15.8%

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

      1. unpow2 15.8%

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

   8. Applied egg-rr 15.8%

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

   if 5.5999999999999999e81 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 88.9%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot \left({y}^{3} \cdot \cos x\right) + y \cdot \cos x}}{y} \]

   6. Taylor expanded in x around 0 65.7%

      \[\leadsto \color{blue}{\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}} \]

   7. Taylor expanded in y around inf 65.7%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot {y}^{3}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 420:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1200:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1200.0)
   (cos x)
   (if (<= y 1.6e+155)
     (/ (+ y (* -0.5 (* y (* x x)))) y)
     (* 0.16666666666666666 (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1200.0) {
		tmp = cos(x);
	} else if (y <= 1.6e+155) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = 0.16666666666666666 * pow(y, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1200.0d0) then
        tmp = cos(x)
    else if (y <= 1.6d+155) then
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    else
        tmp = 0.16666666666666666d0 * (y ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1200.0) {
		tmp = Math.cos(x);
	} else if (y <= 1.6e+155) {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1200.0:
		tmp = math.cos(x)
	elif y <= 1.6e+155:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	else:
		tmp = 0.16666666666666666 * math.pow(y, 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1200.0)
		tmp = cos(x);
	elseif (y <= 1.6e+155)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	else
		tmp = Float64(0.16666666666666666 * (y ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1200.0)
		tmp = cos(x);
	elseif (y <= 1.6e+155)
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	else
		tmp = 0.16666666666666666 * (y ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1200.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.6e+155], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1200

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 1200 < y < 1.60000000000000006e155

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 20.9%

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

      1. unpow2 17.3%

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

   8. Applied egg-rr 20.9%

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

   if 1.60000000000000006e155 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot \left({y}^{3} \cdot \cos x\right) + y \cdot \cos x}}{y} \]

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}} \]

   7. Taylor expanded in y around inf 77.4%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1200:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1400:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1400.0) (cos x) (/ (+ y (* -0.5 (* y (* x x)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1400.0) {
		tmp = cos(x);
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1400.0d0) then
        tmp = cos(x)
    else
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1400.0) {
		tmp = Math.cos(x);
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1400.0:
		tmp = math.cos(x)
	else:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1400.0)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1400.0)
		tmp = cos(x);
	else
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1400.0], N[Cos[x], $MachinePrecision], N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1400

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 1400 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 17.7%

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

      1. unpow2 11.1%

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

   8. Applied egg-rr 17.7%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1400:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 780:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 780.0) 1.0 (/ (+ y (* -0.5 (* y (* x x)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 780.0) {
		tmp = 1.0;
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 780.0d0) then
        tmp = 1.0d0
    else
        tmp = (y + ((-0.5d0) * (y * (x * x)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 780.0) {
		tmp = 1.0;
	} else {
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 780.0:
		tmp = 1.0
	else:
		tmp = (y + (-0.5 * (y * (x * x)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 780.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 780.0)
		tmp = 1.0;
	else
		tmp = (y + (-0.5 * (y * (x * x)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 780.0], 1.0, N[(N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 780

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 65.2%

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

   6. Taylor expanded in x around 0 39.2%

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

   if 780 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 17.7%

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

      1. unpow2 11.1%

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

   8. Applied egg-rr 17.7%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 780:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 840:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 840.0) 1.0 (+ 1.0 (* -0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 840.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((-0.5d0) * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 840.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (-0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 840.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (-0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 840.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 840.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (-0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 840.0], 1.0, N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 840

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 65.2%

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

   6. Taylor expanded in x around 0 39.2%

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

   if 840 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 3.1%

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

   6. Taylor expanded in x around 0 11.1%

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

      1. unpow2 11.1%

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

   8. Applied egg-rr 11.1%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 840:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.2% 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0 51.4%

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

6. Taylor expanded in x around 0 31.1%

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

7. Final simplification 31.1%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))