Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 4

Speedup: 1.0×

• 48.6% 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. Final simplification 100.0%

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

Alternative 2: 69.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 14500000000.0) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 14500000000.0) {
		tmp = cos(x);
	} else {
		tmp = 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 <= 14500000000.0d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 14500000000.0) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 14500000000.0:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 14500000000.0)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 14500000000.0)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 14500000000.0], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e10

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 68.0%

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

   if 1.45e10 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in x around 0 80.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 80.3%

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

      2. associate-/l* 80.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{e^{y} - \frac{1}{e^{y}}}}} \]

      3. *-rgt-identity 80.3%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{y \cdot 1}}{e^{y} - \frac{1}{e^{y}}}} \]

      4. associate-/l* 80.3%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{1}}}} \]

      5. metadata-eval 80.3%

         \[\leadsto \frac{0.5}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{\frac{2}{2}}}}} \]

      6. associate-/l* 80.3%

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

      7. associate-*l/ 80.3%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2} \cdot 2}}} \]

      8. rec-exp 80.3%

         \[\leadsto \frac{0.5}{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2} \cdot 2}} \]

      9. sinh-def 80.3%

         \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\sinh y} \cdot 2}} \]

      10. associate-/l/ 80.3%

          \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{y}{2}}{\sinh y}}} \]

   4. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}} \]
   5. Step-by-step derivation

      1. add-log-exp 80.3%

         \[\leadsto \color{blue}{\log \left(e^{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}}\right)} \]

      2. *-un-lft-identity 80.3%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}}\right)} \]

      3. log-prod 80.3%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}}\right)} \]

      4. metadata-eval 80.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}}\right) \]

      5. div-inv 80.3%

         \[\leadsto 0 + \log \left(e^{\color{blue}{0.5 \cdot \frac{1}{\frac{\frac{y}{2}}{\sinh y}}}}\right) \]

      6. add-log-exp 80.3%

         \[\leadsto 0 + \color{blue}{0.5 \cdot \frac{1}{\frac{\frac{y}{2}}{\sinh y}}} \]

      7. clear-num 80.3%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\frac{\sinh y}{\frac{y}{2}}} \]

      8. div-inv 80.3%

         \[\leadsto 0 + 0.5 \cdot \color{blue}{\left(\sinh y \cdot \frac{1}{\frac{y}{2}}\right)} \]

      9. clear-num 80.3%

         \[\leadsto 0 + 0.5 \cdot \left(\sinh y \cdot \color{blue}{\frac{2}{y}}\right) \]

   6. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{0 + 0.5 \cdot \left(\sinh y \cdot \frac{2}{y}\right)} \]
   7. Step-by-step derivation

      1. +-lft-identity 80.3%

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

      2. associate-*r/ 80.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\sinh y \cdot 2}{y}} \]

      3. *-commutative 80.3%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{2 \cdot \sinh y}}{y} \]

      4. associate-*r/ 80.3%

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

      5. associate-*r* 80.3%

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

      6. metadata-eval 80.3%

         \[\leadsto \frac{\color{blue}{1} \cdot \sinh y}{y} \]

      7. *-lft-identity 80.3%

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

   8. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 3: 52.1% 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. Taylor expanded in y around 0 51.3%

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

3. Final simplification 51.3%

   \[\leadsto \cos x \]

Alternative 4: 28.9% 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. Taylor expanded in x around 0 42.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
3. Step-by-step derivation

   1. associate-*r/ 42.3%

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

   2. associate-/l* 42.3%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{e^{y} - \frac{1}{e^{y}}}}} \]

   3. *-rgt-identity 42.3%

      \[\leadsto \frac{0.5}{\frac{\color{blue}{y \cdot 1}}{e^{y} - \frac{1}{e^{y}}}} \]

   4. associate-/l* 42.3%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{1}}}} \]

   5. metadata-eval 42.3%

      \[\leadsto \frac{0.5}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{\frac{2}{2}}}}} \]

   6. associate-/l* 42.3%

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

   7. associate-*l/ 42.3%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2} \cdot 2}}} \]

   8. rec-exp 42.4%

      \[\leadsto \frac{0.5}{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2} \cdot 2}} \]

   9. sinh-def 65.2%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\sinh y} \cdot 2}} \]

   10. associate-/l/ 65.2%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{y}{2}}{\sinh y}}} \]

4. Simplified 65.2%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{y}{2}}{\sinh y}}} \]

5. Taylor expanded in y around 0 27.3%

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

6. Final simplification 27.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))