Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 6

Speedup: 1.0×

• 49.2% 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: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 - {x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e+29) (cos x) (* 0.5 (- 2.0 (pow x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+29) {
		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 <= 4.5d+29) 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 <= 4.5e+29) {
		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 <= 4.5e+29:
		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 <= 4.5e+29)
		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 <= 4.5e+29)
		tmp = cos(x);
	else
		tmp = 0.5 * (2.0 - (x ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e+29], 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 < 4.5000000000000002e29

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.7%

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

   if 4.5000000000000002e29 < 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 9.3%

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

      1. mul-1-neg 9.3%

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

      2. unsub-neg 9.3%

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

   9. Simplified 9.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 - {x}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \left(-0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e+29) (cos x) (* (pow x 2.0) (- 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e+29) {
		tmp = cos(x);
	} else {
		tmp = pow(x, 2.0) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d+29) then
        tmp = cos(x)
    else
        tmp = (x ** 2.0d0) * -0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e+29) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.pow(x, 2.0) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e+29:
		tmp = math.cos(x)
	else:
		tmp = math.pow(x, 2.0) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e+29)
		tmp = cos(x);
	else
		tmp = Float64((x ^ 2.0) * Float64(-0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e+29)
		tmp = cos(x);
	else
		tmp = (x ^ 2.0) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e+29], N[Cos[x], $MachinePrecision], N[(N[Power[x, 2.0], $MachinePrecision] * (-0.5)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.0000000000000005e29

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.7%

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

   if 9.0000000000000005e29 < 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 9.3%

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

      1. mul-1-neg 9.3%

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

      2. unsub-neg 9.3%

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

   9. Simplified 9.3%

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

   10. Taylor expanded in x around inf 8.3%

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

       1. neg-mul-1 8.3%

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

   12. Simplified 8.3%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \left(-0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% 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 49.2%

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

4. Final simplification 49.2%

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

Alternative 6: 29.5% 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 55.0%

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

   1. *-commutative 55.0%

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

   2. associate-/l* 55.0%

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

   3. rec-exp 55.0%

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

5. Simplified 55.0%

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

6. Taylor expanded in y around 0 49.2%

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

7. Taylor expanded in x around 0 27.4%

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

8. Final simplification 27.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))