Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 5

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

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

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. Add Preprocessing

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.01:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sinh y) y) 1.01) (cos x) (* (sinh y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.01) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sinh(y) / y) <= 1.01d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sinh(y) / y) <= 1.01) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) / y) <= 1.01:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.01)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) / y) <= 1.01)
		tmp = cos(x);
	else
		tmp = sinh(y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], 1.01], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.1%

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

   if 1.01000000000000001 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

      6. *-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \frac{\sinh y}{y} \cdot \cos x} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 76.2%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.01:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.01) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.01) {
		tmp = cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 1.01d0) then
        tmp = cos(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 1.01) {
		tmp = Math.cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.01:
		tmp = math.cos(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.01)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.01)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 1.01], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.1%

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

   if 1.01000000000000001 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

      6. *-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \frac{\sinh y}{y} \cdot \cos x} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 76.2%

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

   8. Taylor expanded in y around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*l/ 76.2%

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

      3. *-rgt-identity 76.2%

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

      4. metadata-eval 76.2%

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

      5. associate-/l* 76.2%

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

      6. associate-*l/ 76.2%

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

      7. rec-exp 76.2%

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

      8. sinh-def 76.2%

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

      9. *-commutative 76.2%

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

      10. metadata-eval 76.2%

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

      11. associate-/l* 76.2%

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

      12. *-rgt-identity 76.2%

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

      13. *-commutative 76.2%

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

      14. associate-/l* 76.2%

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

      15. metadata-eval 76.2%

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

      16. *-rgt-identity 76.2%

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

   10. Simplified 76.2%

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

Alternative 4: 51.3% 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 53.5%

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

Alternative 5: 28.8% 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. add-log-exp 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. log-prod 99.8%

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

   4. metadata-eval 99.8%

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

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\cos x \cdot \frac{\sinh y}{y}} \]

   6. *-commutative 100.0%

      \[\leadsto 0 + \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \frac{\sinh y}{y} \cdot \cos x} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*l/ 99.9%

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

   3. associate-/l* 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in y around 0 53.4%

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

8. Taylor expanded in x around 0 28.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))