Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 6

Speedup: 1.0×

• 49.3% 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: 62.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00085:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00085) (cos x) (/ (+ y (* 0.16666666666666666 (pow y 3.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00085) {
		tmp = cos(x);
	} 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 <= 0.00085d0) then
        tmp = cos(x)
    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 <= 0.00085) {
		tmp = Math.cos(x);
	} else {
		tmp = (y + (0.16666666666666666 * Math.pow(y, 3.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00085:
		tmp = math.cos(x)
	else:
		tmp = (y + (0.16666666666666666 * math.pow(y, 3.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00085)
		tmp = cos(x);
	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 <= 0.00085)
		tmp = cos(x);
	else
		tmp = (y + (0.16666666666666666 * (y ^ 3.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00085], N[Cos[x], $MachinePrecision], N[(N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.49999999999999953e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.6%

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

   if 8.49999999999999953e-4 < 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 67.5%

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

   6. Taylor expanded in x around 0 45.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00085:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.16666666666666666 \cdot {y}^{3}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00055:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00055) (cos x) (+ 1.0 (* 0.16666666666666666 (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00055) {
		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 <= 0.00055d0) 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 <= 0.00055) {
		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 <= 0.00055:
		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 <= 0.00055)
		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 <= 0.00055)
		tmp = cos(x);
	else
		tmp = 1.0 + (0.16666666666666666 * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00055], 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 < 5.50000000000000033e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 67.6%

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

   if 5.50000000000000033e-4 < 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 67.5%

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

   6. Taylor expanded in x around 0 45.2%

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

   7. Taylor expanded in y around 0 34.3%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00055:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot {y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.85 \cdot 10^{+21}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.85e+21) (cos x) (* 0.16666666666666666 (pow y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.85e+21) {
		tmp = cos(x);
	} 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 <= 3.85d+21) then
        tmp = cos(x)
    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 <= 3.85e+21) {
		tmp = Math.cos(x);
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.85e+21:
		tmp = math.cos(x)
	else:
		tmp = 0.16666666666666666 * math.pow(y, 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.85e+21)
		tmp = cos(x);
	else
		tmp = Float64(0.16666666666666666 * (y ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.85e+21)
		tmp = cos(x);
	else
		tmp = 0.16666666666666666 * (y ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.85e+21], N[Cos[x], $MachinePrecision], N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.85e21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.5%

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

   if 3.85e21 < 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 73.4%

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

   6. Taylor expanded in x around 0 48.6%

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

   7. Taylor expanded in y around inf 36.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.85 \cdot 10^{+21}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{2}\\ \end{array} \]
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 50.7%

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

4. Final simplification 50.7%

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

Alternative 6: 28.6% 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 50.6%

   \[\leadsto \frac{\color{blue}{y \cdot \cos x}}{y} \]
6. Step-by-step derivation

   1. *-commutative 50.6%

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

7. Simplified 50.6%

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

8. Taylor expanded in x around 0 26.6%

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

9. Final simplification 26.6%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))