Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 10

Speedup: 1.0×

• 48.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (sin x) (* x t_0))))

C

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

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = x * 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, 1.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

      1. add-log-exp 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. log-prod 95.9%

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

      4. metadata-eval 95.9%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 77.5%

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

   5. Simplified 77.5%

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

   6. Taylor expanded in x around 0 50.0%

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

      1. associate-/r/ 72.5%

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

   8. Applied egg-rr 72.5%

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

4. Final simplification 85.2%

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

Alternative 3: 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. Final simplification 100.0%

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

Alternative 4: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00014:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(\sin x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00014)
   (sin x)
   (if (<= y 6.2e+260)
     (/ x (/ y (sinh y)))
     (* 0.16666666666666666 (* y (* (sin x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00014) {
		tmp = sin(x);
	} else if (y <= 6.2e+260) {
		tmp = x / (y / sinh(y));
	} else {
		tmp = 0.16666666666666666 * (y * (sin(x) * 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 <= 0.00014d0) then
        tmp = sin(x)
    else if (y <= 6.2d+260) then
        tmp = x / (y / sinh(y))
    else
        tmp = 0.16666666666666666d0 * (y * (sin(x) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00014) {
		tmp = Math.sin(x);
	} else if (y <= 6.2e+260) {
		tmp = x / (y / Math.sinh(y));
	} else {
		tmp = 0.16666666666666666 * (y * (Math.sin(x) * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00014:
		tmp = math.sin(x)
	elif y <= 6.2e+260:
		tmp = x / (y / math.sinh(y))
	else:
		tmp = 0.16666666666666666 * (y * (math.sin(x) * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00014)
		tmp = sin(x);
	elseif (y <= 6.2e+260)
		tmp = Float64(x / Float64(y / sinh(y)));
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(sin(x) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00014)
		tmp = sin(x);
	elseif (y <= 6.2e+260)
		tmp = x / (y / sinh(y));
	else
		tmp = 0.16666666666666666 * (y * (sin(x) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00014], N[Sin[x], $MachinePrecision], If[LessEqual[y, 6.2e+260], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3999999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 63.7%

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

   if 1.3999999999999999e-4 < y < 6.1999999999999996e260

   1. Initial program 99.9%

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

      1. add-log-exp 95.0%

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

      2. *-un-lft-identity 95.0%

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

      3. log-prod 95.0%

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

      4. metadata-eval 95.0%

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

      5. add-log-exp 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around 0 43.2%

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

      1. associate-/r/ 69.1%

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

   8. Applied egg-rr 69.1%

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

      1. *-commutative 69.1%

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

      2. clear-num 69.1%

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

      3. un-div-inv 69.1%

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

   10. Applied egg-rr 69.1%

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

   if 6.1999999999999996e260 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00014:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(\sin x \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00012:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00012)
   (sin x)
   (if (<= y 1.35e+154)
     (/ x (/ y (sinh y)))
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00012) {
		tmp = sin(x);
	} else if (y <= 1.35e+154) {
		tmp = x / (y / sinh(y));
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (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 <= 0.00012d0) then
        tmp = sin(x)
    else if (y <= 1.35d+154) then
        tmp = x / (y / sinh(y))
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00012) {
		tmp = Math.sin(x);
	} else if (y <= 1.35e+154) {
		tmp = x / (y / Math.sinh(y));
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00012:
		tmp = math.sin(x)
	elif y <= 1.35e+154:
		tmp = x / (y / math.sinh(y))
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00012)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = Float64(x / Float64(y / sinh(y)));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00012)
		tmp = sin(x);
	elseif (y <= 1.35e+154)
		tmp = x / (y / sinh(y));
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00012], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.20000000000000003e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 63.7%

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

   if 1.20000000000000003e-4 < y < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. add-log-exp 93.0%

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

      2. *-un-lft-identity 93.0%

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

      3. log-prod 93.0%

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

      4. metadata-eval 93.0%

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

      5. add-log-exp 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in x around 0 51.3%

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

      1. associate-/r/ 68.4%

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

   8. Applied egg-rr 68.4%

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

      1. *-commutative 68.4%

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

      2. clear-num 68.5%

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

      3. un-div-inv 68.5%

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

   10. Applied egg-rr 68.5%

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

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00012:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00026:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.00026) (sin x) (/ x (/ y (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00026) {
		tmp = sin(x);
	} else {
		tmp = x / (y / sinh(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 <= 0.00026d0) then
        tmp = sin(x)
    else
        tmp = x / (y / sinh(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00026) {
		tmp = Math.sin(x);
	} else {
		tmp = x / (y / Math.sinh(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00026:
		tmp = math.sin(x)
	else:
		tmp = x / (y / math.sinh(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00026)
		tmp = sin(x);
	else
		tmp = Float64(x / Float64(y / sinh(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00026)
		tmp = sin(x);
	else
		tmp = x / (y / sinh(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00026], N[Sin[x], $MachinePrecision], N[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.59999999999999977e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 63.7%

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

   if 2.59999999999999977e-4 < y

   1. Initial program 100.0%

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

      1. add-log-exp 95.6%

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

      2. *-un-lft-identity 95.6%

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

      3. log-prod 95.6%

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

      4. metadata-eval 95.6%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 71.2%

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

   5. Simplified 71.2%

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

   6. Taylor expanded in x around 0 41.0%

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

      1. associate-/r/ 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. *-commutative 69.8%

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

      2. clear-num 69.8%

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

      3. un-div-inv 69.8%

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

   10. Applied egg-rr 69.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00026:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sinh y}}\\ \end{array} \]

Alternative 7: 62.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.000102:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.000102) (sin x) (* x (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.000102) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * (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 <= 0.000102d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.000102) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.000102:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.000102)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.000102)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.000102], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.01999999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 63.7%

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

   if 1.01999999999999999e-4 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 41.2%

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

      1. unpow2 41.2%

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

   4. Simplified 41.2%

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

   5. Taylor expanded in x around 0 36.9%

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

      1. unpow2 36.9%

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

   7. Simplified 36.9%

      \[\leadsto \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.000102:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 8: 37.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 98000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 98000.0) x (* 0.16666666666666666 (* x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 98000.0) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (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 <= 98000.0d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 98000.0) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 98000.0:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 98000.0)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 98000.0)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 98000.0], x, N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 98000

   1. Initial program 100.0%

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

      1. add-log-exp 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. log-prod 68.1%

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

      4. metadata-eval 68.1%

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

      5. add-log-exp 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l/ 87.3%

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

   3. Applied egg-rr 87.3%

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

      1. +-lft-identity 87.3%

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

      2. associate-/l* 93.7%

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

   5. Simplified 93.7%

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

   6. Taylor expanded in x around 0 54.0%

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

   7. Taylor expanded in y around 0 33.8%

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

   if 98000 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 42.9%

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

      1. unpow2 42.9%

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

   4. Simplified 42.9%

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

   5. Taylor expanded in y around inf 42.9%

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

      1. unpow2 42.9%

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

      2. associate-*l* 30.7%

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

   7. Simplified 30.7%

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

   8. Taylor expanded in x around 0 38.5%

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

      1. unpow2 38.5%

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

      2. *-commutative 38.5%

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

   10. Simplified 38.5%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 98000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 9: 48.2% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return x * (1.0 + (0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return x * (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0 73.2%

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

   1. unpow2 73.2%

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

4. Simplified 73.2%

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

5. Taylor expanded in x around 0 47.3%

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

   1. unpow2 47.3%

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

7. Simplified 47.3%

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

8. Final simplification 47.3%

   \[\leadsto x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 10: 27.2% 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. Step-by-step derivation

   1. add-log-exp 75.8%

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

   2. *-un-lft-identity 75.8%

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

   3. log-prod 75.8%

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

   4. metadata-eval 75.8%

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

   5. add-log-exp 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*l/ 90.4%

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

3. Applied egg-rr 90.4%

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

   1. +-lft-identity 90.4%

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

   2. associate-/l* 87.8%

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

5. Simplified 87.8%

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

6. Taylor expanded in x around 0 51.0%

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

7. Taylor expanded in y around 0 26.2%

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

8. Final simplification 26.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023256 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))