Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 6

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

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

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. 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: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+47}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{y}{y \cdot \left(2 - {x}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e+47) (cos x) (* 0.5 (/ 1.0 (/ y (* y (- 2.0 (pow x 2.0))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e+47) {
		tmp = cos(x);
	} else {
		tmp = 0.5 * (1.0 / (y / (y * (2.0 - pow(x, 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 <= 3d+47) then
        tmp = cos(x)
    else
        tmp = 0.5d0 * (1.0d0 / (y / (y * (2.0d0 - (x ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e+47) {
		tmp = Math.cos(x);
	} else {
		tmp = 0.5 * (1.0 / (y / (y * (2.0 - Math.pow(x, 2.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e+47:
		tmp = math.cos(x)
	else:
		tmp = 0.5 * (1.0 / (y / (y * (2.0 - math.pow(x, 2.0)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e+47)
		tmp = cos(x);
	else
		tmp = Float64(0.5 * Float64(1.0 / Float64(y / Float64(y * Float64(2.0 - (x ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e+47)
		tmp = cos(x);
	else
		tmp = 0.5 * (1.0 / (y / (y * (2.0 - (x ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e+47], N[Cos[x], $MachinePrecision], N[(0.5 * N[(1.0 / N[(y / N[(y * N[(2.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.0000000000000001e47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.9%

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

   if 3.0000000000000001e47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \cos x}}{y} \]

      2. associate-/l* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{\cos x}{y}\right)} \]

      3. rec-exp 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 3.1%

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

      1. associate-*r/ 3.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(2 \cdot y\right) \cdot \cos x}{y}} \]

      2. clear-num 3.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{y}{\left(2 \cdot y\right) \cdot \cos x}}} \]

   8. Applied egg-rr 3.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{y}{\left(2 \cdot y\right) \cdot \cos x}}} \]

   9. Taylor expanded in x around 0 27.4%

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

       1. associate-*r* 27.4%

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

       2. distribute-rgt-out 27.4%

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

       3. +-commutative 27.4%

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

       4. mul-1-neg 27.4%

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

       5. unsub-neg 27.4%

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

   11. Simplified 27.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+47}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{y}{y \cdot \left(2 - {x}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+87}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 - {x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.1e+87) (cos x) (* 0.5 (- 2.0 (pow x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+87) {
		tmp = cos(x);
	} else {
		tmp = 0.5 * (2.0 - pow(x, 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 <= 5.1d+87) then
        tmp = cos(x)
    else
        tmp = 0.5d0 * (2.0d0 - (x ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+87) {
		tmp = Math.cos(x);
	} else {
		tmp = 0.5 * (2.0 - Math.pow(x, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.1e+87:
		tmp = math.cos(x)
	else:
		tmp = 0.5 * (2.0 - math.pow(x, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.1e+87)
		tmp = cos(x);
	else
		tmp = Float64(0.5 * Float64(2.0 - (x ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.1e+87)
		tmp = cos(x);
	else
		tmp = 0.5 * (2.0 - (x ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.1e+87], N[Cos[x], $MachinePrecision], N[(0.5 * N[(2.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.09999999999999988e87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.2%

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

   if 5.09999999999999988e87 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \cos x}}{y} \]

      2. associate-/l* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{\cos x}{y}\right)} \]

      3. rec-exp 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 3.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 \cdot y\right)} \cdot \frac{\cos x}{y}\right) \]

   7. Taylor expanded in x around 0 18.0%

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

      1. mul-1-neg 18.0%

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

      2. unsub-neg 18.0%

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

   9. Simplified 18.0%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+87}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 - {x}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% 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. Add Preprocessing

3. Taylor expanded in y around 0 51.8%

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

4. Final simplification 51.8%

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

Alternative 6: 27.9% 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 inf 52.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\cos x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
4. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \cos x}}{y} \]

   2. associate-/l* 52.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{y} - \frac{1}{e^{y}}\right) \cdot \frac{\cos x}{y}\right)} \]

   3. rec-exp 52.6%

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

5. Simplified 52.6%

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

6. Taylor expanded in y around 0 51.6%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 \cdot y\right)} \cdot \frac{\cos x}{y}\right) \]

7. Taylor expanded in x around 0 28.1%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

8. Final simplification 28.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))