Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 4

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 78000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 78000.0) (sin x) (* x (/ (sinh y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 78000.0) {
		tmp = sin(x);
	} else {
		tmp = x * (sinh(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 <= 78000.0d0) then
        tmp = sin(x)
    else
        tmp = x * (sinh(y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 78000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 78000.0:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 78000.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 78000.0)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 78000.0], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 78000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.1%

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

   if 78000 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/l* 69.8%

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

      3. associate-*r* 69.8%

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

      4. *-commutative 69.8%

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

      5. associate-*r/ 69.8%

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

      6. *-commutative 69.8%

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

      7. metadata-eval 69.8%

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

      8. associate-/l* 69.8%

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

      9. *-rgt-identity 69.8%

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

      10. rec-exp 69.8%

          \[\leadsto x \cdot \frac{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}{y} \]

      11. sinh-def 69.8%

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

   5. Simplified 69.8%

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

Alternative 3: 51.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 49.0%

   \[\leadsto \color{blue}{\sin x} \]
4. Add Preprocessing

Alternative 4: 27.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

   3. associate-*r* 39.3%

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

   4. *-commutative 39.3%

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

   5. associate-*r/ 39.3%

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

   6. *-commutative 39.3%

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

   7. metadata-eval 39.3%

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

   8. associate-/l* 39.3%

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

   9. *-rgt-identity 39.3%

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

   10. rec-exp 39.3%

       \[\leadsto x \cdot \frac{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}{y} \]

   11. sinh-def 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in y around 0 25.0%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Reproduce

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