Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 7

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

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

FPCore

(FPCore (x y) :precision binary64 (/ (cos x) (/ y (sinh y))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{\cos x}{\frac{y}{\sinh y}} \]
6. Add Preprocessing

Alternative 2: 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 3: 55.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 14:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + y \cdot \left({x}^{2} \cdot -0.5\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 14.0) (cos x) (/ (+ y (* y (* (pow x 2.0) -0.5))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 14.0) {
		tmp = cos(x);
	} else {
		tmp = (y + (y * (pow(x, 2.0) * -0.5))) / 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 <= 14.0d0) then
        tmp = cos(x)
    else
        tmp = (y + (y * ((x ** 2.0d0) * (-0.5d0)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 14.0) {
		tmp = Math.cos(x);
	} else {
		tmp = (y + (y * (Math.pow(x, 2.0) * -0.5))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 14.0:
		tmp = math.cos(x)
	else:
		tmp = (y + (y * (math.pow(x, 2.0) * -0.5))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 14.0)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y + Float64(y * Float64((x ^ 2.0) * -0.5))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 14.0)
		tmp = cos(x);
	else
		tmp = (y + (y * ((x ^ 2.0) * -0.5))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 14.0], N[Cos[x], $MachinePrecision], N[(N[(y + N[(y * N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 59.3%

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

   if 14 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. rec-exp 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 3.1%

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

   7. Taylor expanded in x around 0 11.6%

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

      1. associate-*r* 11.6%

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

      2. *-commutative 11.6%

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

   9. Simplified 11.6%

      \[\leadsto \frac{\color{blue}{y + \left({x}^{2} \cdot -0.5\right) \cdot y}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + y \cdot \left({x}^{2} \cdot -0.5\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 14:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;3 + {x}^{2} \cdot -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 14.0) (cos x) (+ 3.0 (* (pow x 2.0) -1.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 14.0) {
		tmp = cos(x);
	} else {
		tmp = 3.0 + (pow(x, 2.0) * -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 14.0d0) then
        tmp = cos(x)
    else
        tmp = 3.0d0 + ((x ** 2.0d0) * (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 14.0) {
		tmp = Math.cos(x);
	} else {
		tmp = 3.0 + (Math.pow(x, 2.0) * -1.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 14.0:
		tmp = math.cos(x)
	else:
		tmp = 3.0 + (math.pow(x, 2.0) * -1.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 14.0)
		tmp = cos(x);
	else
		tmp = Float64(3.0 + Float64((x ^ 2.0) * -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 14.0)
		tmp = cos(x);
	else
		tmp = 3.0 + ((x ^ 2.0) * -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 14.0], N[Cos[x], $MachinePrecision], N[(3.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 59.3%

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

   if 14 < 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%

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

      2. add-sqr-sqrt 100.0%

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

      3. log-prod 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{\sinh y}{y}}}\right) + \log \left(\sqrt{e^{\frac{\sinh y}{y}}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 3.1%

      \[\leadsto \cos x \cdot \left(2 \cdot \color{blue}{1.5}\right) \]

   9. Taylor expanded in x around 0 11.6%

      \[\leadsto \color{blue}{3 + -1.5 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;3 + {x}^{2} \cdot -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e+27) (cos x) (* (pow x 2.0) -0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+27) {
		tmp = cos(x);
	} else {
		tmp = pow(x, 2.0) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d+27) then
        tmp = cos(x)
    else
        tmp = (x ** 2.0d0) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+27) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.pow(x, 2.0) * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e+27:
		tmp = math.cos(x)
	else:
		tmp = math.pow(x, 2.0) * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e+27)
		tmp = cos(x);
	else
		tmp = Float64((x ^ 2.0) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e+27)
		tmp = cos(x);
	else
		tmp = (x ^ 2.0) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e+27], N[Cos[x], $MachinePrecision], N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999991e27

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 58.0%

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

   if 1.79999999999999991e27 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. rec-exp 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 3.1%

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

   7. Taylor expanded in x around 0 12.4%

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

      1. associate-*r* 12.4%

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

      2. *-commutative 12.4%

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

   9. Simplified 12.4%

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

   10. Taylor expanded in x around inf 11.3%

       \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 47.5%

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

4. Final simplification 47.5%

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

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around inf 57.1%

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

   1. associate-*r/ 57.1%

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

   2. associate-*r* 57.1%

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

   3. rec-exp 57.2%

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

5. Simplified 57.2%

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

6. Taylor expanded in y around 0 47.4%

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

7. Taylor expanded in x around 0 25.4%

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

8. Final simplification 25.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))