Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 6

Speedup: 1.0×

• 49.6% 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: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{\cos x}{y \cdot -0.16666666666666666 + \frac{1}{y}}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= t_0 2.0)
     (/ 1.0 (/ y (/ (cos x) (+ (* y -0.16666666666666666) (/ 1.0 y)))))
     t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = 1.0 / (y / (cos(x) / ((y * -0.16666666666666666) + (1.0 / y))));
	} 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 <= 2.0d0) then
        tmp = 1.0d0 / (y / (cos(x) / ((y * (-0.16666666666666666d0)) + (1.0d0 / y))))
    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 <= 2.0) {
		tmp = 1.0 / (y / (Math.cos(x) / ((y * -0.16666666666666666) + (1.0 / y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 2.0:
		tmp = 1.0 / (y / (math.cos(x) / ((y * -0.16666666666666666) + (1.0 / y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(1.0 / Float64(y / Float64(cos(x) / Float64(Float64(y * -0.16666666666666666) + Float64(1.0 / y)))));
	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 <= 2.0)
		tmp = 1.0 / (y / (cos(x) / ((y * -0.16666666666666666) + (1.0 / y))));
	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, 2.0], N[(1.0 / N[(y / N[(N[Cos[x], $MachinePrecision] / N[(N[(y * -0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cos x \cdot \sinh y}}} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. sinh-def 7.1%

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

      3. associate-*l/ 7.1%

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

      4. sinh-undef 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{\frac{\left(2 \cdot \sinh y\right) \cdot \cos x}{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. associate-/l* 99.6%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{\frac{\cos x}{\frac{2}{2 \cdot \sinh y}}}}} \]

      3. associate-/r* 99.6%

         \[\leadsto \frac{1}{\frac{y}{\frac{\cos x}{\color{blue}{\frac{\frac{2}{2}}{\sinh y}}}}} \]

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 99.2%

      \[\leadsto \frac{1}{\frac{y}{\frac{\cos x}{\color{blue}{-0.16666666666666666 \cdot y + \frac{1}{y}}}}} \]

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

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

      3. *-commutative 100.0%

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

      4. clear-num 100.0%

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

      5. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 76.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 2:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{\cos x}{y \cdot -0.16666666666666666 + \frac{1}{y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 2.0) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		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 <= 2.0:
		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 <= 2.0)
		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 <= 2.0)
		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, 2.0], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 98.7%

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

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

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

      3. *-commutative 100.0%

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

      4. clear-num 100.0%

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

      5. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 76.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 2:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 4: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e+20) (cos x) (* y (* -0.5 (* x (* x (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+20) {
		tmp = cos(x);
	} else {
		tmp = y * (-0.5 * (x * (x * (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 (y <= 1.35d+20) then
        tmp = cos(x)
    else
        tmp = y * ((-0.5d0) * (x * (x * (1.0d0 / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+20) {
		tmp = Math.cos(x);
	} else {
		tmp = y * (-0.5 * (x * (x * (1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e+20:
		tmp = math.cos(x)
	else:
		tmp = y * (-0.5 * (x * (x * (1.0 / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e+20)
		tmp = cos(x);
	else
		tmp = Float64(y * Float64(-0.5 * Float64(x * Float64(x * Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e+20)
		tmp = cos(x);
	else
		tmp = y * (-0.5 * (x * (x * (1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.35e+20], N[Cos[x], $MachinePrecision], N[(y * N[(-0.5 * N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e20

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 67.0%

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

   if 1.35e20 < 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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 20.5%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {x}^{2}}}{y} \cdot y \]
   6. Step-by-step derivation

      1. *-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in x around inf 19.7%

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

      1. div-inv 19.7%

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

      2. unpow2 19.7%

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

      3. associate-*l* 18.5%

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

   10. Applied egg-rr 18.5%

       \[\leadsto \left(-0.5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)}\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5: 31.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.22e+20) 1.0 (* y (* -0.5 (* x (* x (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.22e+20) {
		tmp = 1.0;
	} else {
		tmp = y * (-0.5 * (x * (x * (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 (y <= 1.22d+20) then
        tmp = 1.0d0
    else
        tmp = y * ((-0.5d0) * (x * (x * (1.0d0 / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.22e+20) {
		tmp = 1.0;
	} else {
		tmp = y * (-0.5 * (x * (x * (1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.22e+20:
		tmp = 1.0
	else:
		tmp = y * (-0.5 * (x * (x * (1.0 / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.22e+20)
		tmp = 1.0;
	else
		tmp = Float64(y * Float64(-0.5 * Float64(x * Float64(x * Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.22e+20)
		tmp = 1.0;
	else
		tmp = y * (-0.5 * (x * (x * (1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.22e+20], 1.0, N[(y * N[(-0.5 * N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.22e20

   1. Initial program 100.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 66.9%

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

   5. Taylor expanded in x around 0 35.6%

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

   if 1.22e20 < 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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 20.5%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {x}^{2}}}{y} \cdot y \]
   6. Step-by-step derivation

      1. *-commutative 20.5%

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

   7. Simplified 20.5%

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

   8. Taylor expanded in x around inf 19.7%

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

      1. div-inv 19.7%

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

      2. unpow2 19.7%

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

      3. associate-*l* 18.5%

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

   10. Applied egg-rr 18.5%

       \[\leadsto \left(-0.5 \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)}\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]

Alternative 6: 28.3% 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/ 99.9%

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

   2. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 50.5%

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

5. Taylor expanded in x around 0 27.0%

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

6. Final simplification 27.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))