Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 1.0×

• 49.9% 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: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.015:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\cos x \cdot y\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.015)
   (cos x)
   (if (<= y 1.8e+154) (/ (sinh y) y) (/ (/ (* y (* (cos x) y)) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.015) {
		tmp = cos(x);
	} else if (y <= 1.8e+154) {
		tmp = sinh(y) / y;
	} else {
		tmp = ((y * (cos(x) * y)) / 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 <= 0.015d0) then
        tmp = cos(x)
    else if (y <= 1.8d+154) then
        tmp = sinh(y) / y
    else
        tmp = ((y * (cos(x) * y)) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.015) {
		tmp = Math.cos(x);
	} else if (y <= 1.8e+154) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = ((y * (Math.cos(x) * y)) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.015:
		tmp = math.cos(x)
	elif y <= 1.8e+154:
		tmp = math.sinh(y) / y
	else:
		tmp = ((y * (math.cos(x) * y)) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.015)
		tmp = cos(x);
	elseif (y <= 1.8e+154)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(Float64(Float64(y * Float64(cos(x) * y)) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.015)
		tmp = cos(x);
	elseif (y <= 1.8e+154)
		tmp = sinh(y) / y;
	else
		tmp = ((y * (cos(x) * y)) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.015], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.8e+154], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(y * N[(N[Cos[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.014999999999999999

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 81.6%

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

   if 0.014999999999999999 < y < 1.8e154

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 65.8%

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

   7. Taylor expanded in y around inf 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-/r/ 65.8%

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

      3. metadata-eval 65.8%

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

      4. associate-/l* 65.8%

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

      5. *-commutative 65.8%

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

      6. /-rgt-identity 65.8%

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

      7. associate-/r* 65.8%

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

      8. rec-exp 65.8%

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

      9. sinh-def 65.8%

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

   9. Simplified 65.8%

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

   if 1.8e154 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. associate-*r/ 3.1%

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

   8. Applied egg-rr 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. *-commutative 3.1%

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

      3. lft-mult-inverse 3.1%

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

      4. associate-*l* 3.1%

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

      5. div-inv 3.1%

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

      6. associate-*l/ 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \frac{\color{blue}{\frac{\left(y \cdot \cos x\right) \cdot y}{y}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.015:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\cos x \cdot y\right)}{y}}{y}\\ \end{array} \]

Alternative 3: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00108:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.00108) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00108) {
		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 <= 0.00108d0) 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 <= 0.00108) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00108:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00108)
		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 <= 0.00108)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00108], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00108000000000000001

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 81.6%

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

   if 0.00108000000000000001 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 68.8%

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

   7. Taylor expanded in y around inf 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-/r/ 68.8%

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

      3. metadata-eval 68.8%

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

      4. associate-/l* 68.8%

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

      5. *-commutative 68.8%

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

      6. /-rgt-identity 68.8%

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

      7. associate-/r* 68.8%

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

      8. rec-exp 68.8%

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

      9. sinh-def 68.8%

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

   9. Simplified 68.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00108:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 4: 50.6% 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. Step-by-step derivation

   1. associate-*r/ 100.0%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 58.0%

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

5. Final simplification 58.0%

   \[\leadsto \cos x \]

Alternative 5: 28.2% 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. Step-by-step derivation

   1. associate-*r/ 100.0%

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

   2. associate-/l* 100.0%

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

3. Simplified 100.0%

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

   1. associate-/r/ 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 60.0%

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

7. Taylor expanded in y around 0 30.9%

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

8. Final simplification 30.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))