Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 6

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 61.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4600000000000:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4600000000000.0)
   (sin x)
   (if (<= y 4.2e+108)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (* 0.16666666666666666 (* x (pow y 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4600000000000.0) {
		tmp = sin(x);
	} else if (y <= 4.2e+108) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = 0.16666666666666666 * (x * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4600000000000.0d0) then
        tmp = sin(x)
    else if (y <= 4.2d+108) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = 0.16666666666666666d0 * (x * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4600000000000.0) {
		tmp = Math.sin(x);
	} else if (y <= 4.2e+108) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = 0.16666666666666666 * (x * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4600000000000.0:
		tmp = math.sin(x)
	elif y <= 4.2e+108:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = 0.16666666666666666 * (x * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4600000000000.0)
		tmp = sin(x);
	elseif (y <= 4.2e+108)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(x * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4600000000000.0)
		tmp = sin(x);
	elseif (y <= 4.2e+108)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4600000000000.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 4.2e+108], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.6e12

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.8%

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

   if 4.6e12 < y < 4.20000000000000019e108

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ 95.5%

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

      2. associate-*l/ 100.0%

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

      3. div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 2.7%

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

   8. Taylor expanded in x around 0 28.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in 28.9%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 28.9%

         \[\leadsto \color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 28.9%

         \[\leadsto x + \color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 28.9%

         \[\leadsto x + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      5. unpow3 28.9%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   10. Simplified 28.9%

       \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 4.20000000000000019e108 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 80.8%

      \[\leadsto \color{blue}{\sin x + 0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 80.8%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\sin x + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   6. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

   7. Taylor expanded in y around inf 67.5%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e+42) (sin x) (* 0.16666666666666666 (* x (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e+42) {
		tmp = sin(x);
	} else {
		tmp = 0.16666666666666666 * (x * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d+42) then
        tmp = sin(x)
    else
        tmp = 0.16666666666666666d0 * (x * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e+42) {
		tmp = Math.sin(x);
	} else {
		tmp = 0.16666666666666666 * (x * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e+42:
		tmp = math.sin(x)
	else:
		tmp = 0.16666666666666666 * (x * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e+42)
		tmp = sin(x);
	else
		tmp = Float64(0.16666666666666666 * Float64(x * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e+42)
		tmp = sin(x);
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e+42], N[Sin[x], $MachinePrecision], N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999995e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.0%

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

   if 1.29999999999999995e42 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 58.5%

      \[\leadsto \color{blue}{\sin x + 0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 58.5%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\sin x + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   6. Taylor expanded in x around 0 50.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

   7. Taylor expanded in y around inf 50.4%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+89}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e+89) (sin x) (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+89) {
		tmp = sin(x);
	} else {
		tmp = (x * 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 (y <= 6.2d+89) then
        tmp = sin(x)
    else
        tmp = (x * y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+89) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.2e+89:
		tmp = math.sin(x)
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e+89)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e+89)
		tmp = sin(x);
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e+89], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2e89

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 58.0%

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

   if 6.2e89 < y

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ 61.4%

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

      2. associate-*l/ 100.0%

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

      3. div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 10.8%

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

Alternative 5: 28.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4.2e+60) x (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.2e+60) {
		tmp = x;
	} else {
		tmp = (x * 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 (x <= 4.2d+60) then
        tmp = x
    else
        tmp = (x * y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.2e+60) {
		tmp = x;
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.2e+60:
		tmp = x
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.2e+60)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.2e+60)
		tmp = x;
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.2e+60], x, N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000002e60

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 75.4%

      \[\leadsto \color{blue}{\sin x + 0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 75.4%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\sin x + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

   6. Taylor expanded in x around 0 50.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

   7. Taylor expanded in y around 0 26.7%

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

   if 4.2000000000000002e60 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 57.9%

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

   8. Taylor expanded in x around 0 14.6%

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

Alternative 6: 26.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

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

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 75.1%

   \[\leadsto \color{blue}{\sin x + 0.16666666666666666 \cdot \left({y}^{2} \cdot \sin x\right)} \]
4. Step-by-step derivation

   1. associate-*r* 75.1%

      \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

5. Simplified 75.1%

   \[\leadsto \color{blue}{\sin x + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

6. Taylor expanded in x around 0 42.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]

7. Taylor expanded in y around 0 21.6%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))