Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

• 49.7% 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: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.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 67.2%

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

4. Final simplification 84.8%

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

Alternative 3: 63.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 20500000:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;-0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 20500000.0)
   (cos x)
   (if (<= y 1.45e+96)
     (* -0.5 (pow x 2.0))
     (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 20500000.0) {
		tmp = cos(x);
	} else if (y <= 1.45e+96) {
		tmp = -0.5 * pow(x, 2.0);
	} else {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * 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 <= 20500000.0d0) then
        tmp = cos(x)
    else if (y <= 1.45d+96) then
        tmp = (-0.5d0) * (x ** 2.0d0)
    else
        tmp = (y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 20500000.0) {
		tmp = Math.cos(x);
	} else if (y <= 1.45e+96) {
		tmp = -0.5 * Math.pow(x, 2.0);
	} else {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 20500000.0:
		tmp = math.cos(x)
	elif y <= 1.45e+96:
		tmp = -0.5 * math.pow(x, 2.0)
	else:
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 20500000.0)
		tmp = cos(x);
	elseif (y <= 1.45e+96)
		tmp = Float64(-0.5 * (x ^ 2.0));
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 20500000.0)
		tmp = cos(x);
	elseif (y <= 1.45e+96)
		tmp = -0.5 * (x ^ 2.0);
	else
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 20500000.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.45e+96], N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.05e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 89.5%

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

   4. Taylor expanded in y around 0 69.7%

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

   if 2.05e7 < y < 1.44999999999999989e96

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 4.1%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 29.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]

   6. Taylor expanded in x around inf 29.3%

      \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]

   if 1.44999999999999989e96 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.7%

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

   4. Taylor expanded in y around 0 67.0%

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

      1. unpow2 67.0%

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

   6. Applied egg-rr 67.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 20500000:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;-0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+72}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e+72)
   (cos x)
   (/ (* y (+ 1.0 (* 0.16666666666666666 (* y y)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+72) {
		tmp = cos(x);
	} else {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * 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 <= 1.02d+72) then
        tmp = cos(x)
    else
        tmp = (y * (1.0d0 + (0.16666666666666666d0 * (y * y)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+72) {
		tmp = Math.cos(x);
	} else {
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.02e+72:
		tmp = math.cos(x)
	else:
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.02e+72)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e+72)
		tmp = cos(x);
	else
		tmp = (y * (1.0 + (0.16666666666666666 * (y * y)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.02e+72], N[Cos[x], $MachinePrecision], N[(N[(y * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 83.3%

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

   4. Taylor expanded in y around 0 64.8%

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

   if 1.02e72 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.7%

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

   4. Taylor expanded in y around 0 58.7%

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

      1. unpow2 58.7%

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

   6. Applied egg-rr 58.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+72}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 59.2%

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

4. Taylor expanded in y around 0 48.4%

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

   1. unpow2 48.4%

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

6. Applied egg-rr 48.4%

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

7. Final simplification 48.4%

   \[\leadsto \frac{y \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{y} \]
8. Add Preprocessing

Alternative 6: 28.7% 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 83.1%

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

4. Taylor expanded in y around 0 55.6%

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

5. Taylor expanded in x around 0 29.7%

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

6. Final simplification 29.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))