Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 8

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

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

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: 84.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 27500 \lor \neg \left(y \leq 3.6 \cdot 10^{+148}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 27500.0) (not (<= y 3.6e+148)))
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (* (/ (sinh y) y) (+ 1.0 (* (* x x) -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 27500.0) || !(y <= 3.6e+148)) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = (sinh(y) / y) * (1.0 + ((x * x) * -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 <= 27500.0d0) .or. (.not. (y <= 3.6d+148))) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else
        tmp = (sinh(y) / y) * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 27500.0) || !(y <= 3.6e+148)) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else {
		tmp = (Math.sinh(y) / y) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 27500.0) or not (y <= 3.6e+148):
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	else:
		tmp = (math.sinh(y) / y) * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 27500.0) || !(y <= 3.6e+148))
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(Float64(sinh(y) / y) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 27500.0) || ~((y <= 3.6e+148)))
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	else
		tmp = (sinh(y) / y) * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 27500.0], N[Not[LessEqual[y, 3.6e+148]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 27500 or 3.60000000000000006e148 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 86.9%

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

      1. unpow2 86.9%

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

   4. Simplified 86.9%

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

   if 27500 < y < 3.60000000000000006e148

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

      1. *-commutative 11.9%

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

      2. unpow2 11.9%

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

   4. Simplified 66.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 27500 \lor \neg \left(y \leq 3.6 \cdot 10^{+148}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 3: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 27500 \lor \neg \left(y \leq 3.6 \cdot 10^{+148}\right):\\ \;\;\;\;\cos x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))
   (if (or (<= y 27500.0) (not (<= y 3.6e+148)))
     (* (cos x) t_0)
     (* t_0 (+ 1.0 (* (* x x) -0.5))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if ((y <= 27500.0) || !(y <= 3.6e+148)) {
		tmp = cos(x) * t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	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 = 1.0d0 + (0.16666666666666666d0 * (y * y))
    if ((y <= 27500.0d0) .or. (.not. (y <= 3.6d+148))) then
        tmp = cos(x) * t_0
    else
        tmp = t_0 * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.16666666666666666 * (y * y));
	double tmp;
	if ((y <= 27500.0) || !(y <= 3.6e+148)) {
		tmp = Math.cos(x) * t_0;
	} else {
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.16666666666666666 * (y * y))
	tmp = 0
	if (y <= 27500.0) or not (y <= 3.6e+148):
		tmp = math.cos(x) * t_0
	else:
		tmp = t_0 * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if ((y <= 27500.0) || !(y <= 3.6e+148))
		tmp = Float64(cos(x) * t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.16666666666666666 * (y * y));
	tmp = 0.0;
	if ((y <= 27500.0) || ~((y <= 3.6e+148)))
		tmp = cos(x) * t_0;
	else
		tmp = t_0 * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, 27500.0], N[Not[LessEqual[y, 3.6e+148]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 27500 or 3.60000000000000006e148 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 86.9%

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

      1. unpow2 86.9%

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

   4. Simplified 86.9%

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

   if 27500 < y < 3.60000000000000006e148

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 5.0%

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

      1. unpow2 5.0%

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

   4. Simplified 5.0%

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

   5. Taylor expanded in x around 0 19.3%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 11.9%

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

      2. unpow2 11.9%

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

   7. Simplified 19.3%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 27500 \lor \neg \left(y \leq 3.6 \cdot 10^{+148}\right):\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 4: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 14800000000:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+196} \lor \neg \left(y \leq 1.4 \cdot 10^{+226}\right):\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 14800000000.0)
   (cos x)
   (if (or (<= y 3e+196) (not (<= y 1.4e+226)))
     (* (+ 1.0 (* 0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) -0.5)))
     (+ 1.0 (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 14800000000.0) {
		tmp = cos(x);
	} else if ((y <= 3e+196) || !(y <= 1.4e+226)) {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 14800000000.0d0) then
        tmp = cos(x)
    else if ((y <= 3d+196) .or. (.not. (y <= 1.4d+226))) then
        tmp = (1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((x * x) * (-0.5d0)))
    else
        tmp = 1.0d0 + (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 14800000000.0) {
		tmp = Math.cos(x);
	} else if ((y <= 3e+196) || !(y <= 1.4e+226)) {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	} else {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 14800000000.0:
		tmp = math.cos(x)
	elif (y <= 3e+196) or not (y <= 1.4e+226):
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5))
	else:
		tmp = 1.0 + (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 14800000000.0)
		tmp = cos(x);
	elseif ((y <= 3e+196) || !(y <= 1.4e+226))
		tmp = Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 14800000000.0)
		tmp = cos(x);
	elseif ((y <= 3e+196) || ~((y <= 1.4e+226)))
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	else
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 14800000000.0], N[Cos[x], $MachinePrecision], If[Or[LessEqual[y, 3e+196], N[Not[LessEqual[y, 1.4e+226]], $MachinePrecision]], N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.48e10

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.6%

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

   if 1.48e10 < y < 2.9999999999999999e196 or 1.4000000000000001e226 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 52.6%

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

      1. unpow2 52.6%

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

   4. Simplified 52.6%

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

   5. Taylor expanded in x around 0 56.1%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 18.2%

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

      2. unpow2 18.2%

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

   7. Simplified 56.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   if 2.9999999999999999e196 < y < 1.4000000000000001e226

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 0.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 0.1%

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

      2. unpow2 0.1%

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

   7. Simplified 0.0%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{2} + 1} \]

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]
   11. Step-by-step derivation

       1. fma-udef 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14800000000:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+196} \lor \neg \left(y \leq 1.4 \cdot 10^{+226}\right):\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 5: 49.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0037 \lor \neg \left(y \leq 4 \cdot 10^{+197}\right) \land y \leq 1.8 \cdot 10^{+226}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 0.0037) (and (not (<= y 4e+197)) (<= y 1.8e+226)))
   (+ 1.0 (* y (* y 0.16666666666666666)))
   (* (+ 1.0 (* 0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 0.0037) || (!(y <= 4e+197) && (y <= 1.8e+226))) {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -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 <= 0.0037d0) .or. (.not. (y <= 4d+197)) .and. (y <= 1.8d+226)) then
        tmp = 1.0d0 + (y * (y * 0.16666666666666666d0))
    else
        tmp = (1.0d0 + (0.16666666666666666d0 * (y * y))) * (1.0d0 + ((x * x) * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 0.0037) || (!(y <= 4e+197) && (y <= 1.8e+226))) {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	} else {
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 0.0037) or (not (y <= 4e+197) and (y <= 1.8e+226)):
		tmp = 1.0 + (y * (y * 0.16666666666666666))
	else:
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 0.0037) || (!(y <= 4e+197) && (y <= 1.8e+226)))
		tmp = Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)));
	else
		tmp = Float64(Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 0.0037) || (~((y <= 4e+197)) && (y <= 1.8e+226)))
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	else
		tmp = (1.0 + (0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 0.0037], And[N[Not[LessEqual[y, 4e+197]], $MachinePrecision], LessEqual[y, 1.8e+226]]], N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0037000000000000002 or 3.9999999999999998e197 < y < 1.7999999999999999e226

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.9%

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

      1. unpow2 85.9%

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

   4. Simplified 85.9%

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

   5. Taylor expanded in x around 0 41.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 31.3%

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

      2. unpow2 31.3%

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

   7. Simplified 41.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0 46.4%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-commutative 46.4%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{2} + 1} \]

      2. unpow2 46.4%

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

      3. fma-udef 46.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   10. Simplified 46.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]
   11. Step-by-step derivation

       1. fma-udef 46.4%

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

       2. *-commutative 46.4%

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

       3. associate-*r* 46.4%

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

       4. *-commutative 46.4%

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

   12. Applied egg-rr 46.4%

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

   if 0.0037000000000000002 < y < 3.9999999999999998e197 or 1.7999999999999999e226 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 52.1%

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

      1. unpow2 52.1%

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

   4. Simplified 52.1%

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

   5. Taylor expanded in x around 0 54.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 18.2%

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

      2. unpow2 18.2%

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

   7. Simplified 54.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0037 \lor \neg \left(y \leq 4 \cdot 10^{+197}\right) \land y \leq 1.8 \cdot 10^{+226}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 6: 32.3% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0026:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0026) 1.0 (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0026) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x * x) * -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 <= 0.0026d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0026) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0026:
		tmp = 1.0
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0026)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0026)
		tmp = 1.0;
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0026], 1.0, N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0025999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 69.2%

      \[\leadsto \color{blue}{\cos x} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 69.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \]

      2. expm1-udef 68.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x\right)} - 1} \]

      3. log1p-udef 68.9%

         \[\leadsto e^{\color{blue}{\log \left(1 + \cos x\right)}} - 1 \]

      4. rem-exp-log 68.9%

         \[\leadsto \color{blue}{\left(1 + \cos x\right)} - 1 \]

   4. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(1 + \cos x\right) - 1} \]
   5. Step-by-step derivation

      1. associate--l+ 68.9%

         \[\leadsto \color{blue}{1 + \left(\cos x - 1\right)} \]

      2. +-commutative 68.9%

         \[\leadsto \color{blue}{\left(\cos x - 1\right) + 1} \]

      3. sub-neg 68.9%

         \[\leadsto \color{blue}{\left(\cos x + \left(-1\right)\right)} + 1 \]

      4. metadata-eval 68.9%

         \[\leadsto \left(\cos x + \color{blue}{-1}\right) + 1 \]

   6. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(\cos x + -1\right) + 1} \]

   7. Taylor expanded in x around 0 32.3%

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

   if 0.0025999999999999999 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 4.2%

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

   3. Taylor expanded in x around 0 17.4%

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

      1. *-commutative 17.4%

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

      2. unpow2 17.4%

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

   5. Simplified 17.4%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0026:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 7: 47.8% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+235}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.8e+235)
   (+ 1.0 (* y (* y 0.16666666666666666)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e+235) {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * -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 (x <= 6.8d+235) then
        tmp = 1.0d0 + (y * (y * 0.16666666666666666d0))
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.8e+235) {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e+235:
		tmp = 1.0 + (y * (y * 0.16666666666666666))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e+235)
		tmp = Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e+235)
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e+235], N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999991e235

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.7%

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

      1. unpow2 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in x around 0 45.8%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 27.6%

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

      2. unpow2 27.6%

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

   7. Simplified 45.8%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0 45.4%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-commutative 45.4%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{2} + 1} \]

      2. unpow2 45.4%

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

      3. fma-udef 45.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]

   10. Simplified 45.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \]
   11. Step-by-step derivation

       1. fma-udef 45.4%

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

       2. *-commutative 45.4%

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

       3. associate-*r* 45.4%

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

       4. *-commutative 45.4%

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

   12. Applied egg-rr 45.4%

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

   if 6.79999999999999991e235 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 61.2%

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

   3. Taylor expanded in x around 0 34.4%

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

      1. *-commutative 34.4%

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

      2. unpow2 34.4%

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

   5. Simplified 34.4%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+235}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 8: 29.1% 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. Taylor expanded in y around 0 52.0%

   \[\leadsto \color{blue}{\cos x} \]
3. Step-by-step derivation

   1. expm1-log1p-u 51.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \]

   2. expm1-udef 51.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x\right)} - 1} \]

   3. log1p-udef 51.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + \cos x\right)}} - 1 \]

   4. rem-exp-log 51.7%

      \[\leadsto \color{blue}{\left(1 + \cos x\right)} - 1 \]

4. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{\left(1 + \cos x\right) - 1} \]
5. Step-by-step derivation

   1. associate--l+ 51.7%

      \[\leadsto \color{blue}{1 + \left(\cos x - 1\right)} \]

   2. +-commutative 51.7%

      \[\leadsto \color{blue}{\left(\cos x - 1\right) + 1} \]

   3. sub-neg 51.7%

      \[\leadsto \color{blue}{\left(\cos x + \left(-1\right)\right)} + 1 \]

   4. metadata-eval 51.7%

      \[\leadsto \left(\cos x + \color{blue}{-1}\right) + 1 \]

6. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{\left(\cos x + -1\right) + 1} \]

7. Taylor expanded in x around 0 24.5%

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

8. Final simplification 24.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))