Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 5

Speedup: 1.0×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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. Step-by-step derivation

   1. associate-*r/ 99.9%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\cos x}{\frac{y}{\sinh y}} \]
6. 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. Final simplification 100.0%

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

Alternative 3: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+30) (cos x) (+ 1.0 (* 0.16666666666666666 (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+30) {
		tmp = cos(x);
	} 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 <= 7.2d+30) then
        tmp = cos(x)
    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 <= 7.2e+30) {
		tmp = Math.cos(x);
	} else {
		tmp = 1.0 + (0.16666666666666666 * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+30:
		tmp = math.cos(x)
	else:
		tmp = 1.0 + (0.16666666666666666 * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+30)
		tmp = cos(x);
	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 <= 7.2e+30)
		tmp = cos(x);
	else
		tmp = 1.0 + (0.16666666666666666 * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+30], N[Cos[x], $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000004e30

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 65.0%

      \[\leadsto \frac{\cos x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]

   6. Taylor expanded in y around 0 65.3%

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

   if 7.2000000000000004e30 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 1.4%

      \[\leadsto \frac{\cos x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]

   6. Taylor expanded in x around 0 1.4%

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

   7. Taylor expanded in y around 0 46.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+30}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cos(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cos[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 99.9%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 47.9%

   \[\leadsto \frac{\cos x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]

6. Taylor expanded in y around 0 48.6%

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

7. Final simplification 48.6%

   \[\leadsto \cos x \]
8. Add Preprocessing

Alternative 5: 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. Step-by-step derivation

   1. associate-*r/ 99.9%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around 0 47.9%

   \[\leadsto \frac{\cos x}{\color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}}} \]

6. Taylor expanded in x around 0 26.9%

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

7. Taylor expanded in y around 0 27.4%

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

8. Final simplification 27.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))