Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 6

Speedup: 1.0×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0036:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\cos x \cdot y\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0036)
   (cos x)
   (if (<= y 1.05e+148) (/ (sinh y) y) (/ (/ (* y (* (cos x) y)) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0036) {
		tmp = cos(x);
	} else if (y <= 1.05e+148) {
		tmp = sinh(y) / y;
	} else {
		tmp = ((y * (cos(x) * y)) / 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 <= 0.0036d0) then
        tmp = cos(x)
    else if (y <= 1.05d+148) then
        tmp = sinh(y) / y
    else
        tmp = ((y * (cos(x) * y)) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0036) {
		tmp = Math.cos(x);
	} else if (y <= 1.05e+148) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = ((y * (Math.cos(x) * y)) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0036:
		tmp = math.cos(x)
	elif y <= 1.05e+148:
		tmp = math.sinh(y) / y
	else:
		tmp = ((y * (math.cos(x) * y)) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0036)
		tmp = cos(x);
	elseif (y <= 1.05e+148)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(Float64(Float64(y * Float64(cos(x) * y)) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0036)
		tmp = cos(x);
	elseif (y <= 1.05e+148)
		tmp = sinh(y) / y;
	else
		tmp = ((y * (cos(x) * y)) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0036], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.05e+148], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(y * N[(N[Cos[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0035999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if 0.0035999999999999999 < y < 1.04999999999999999e148

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 71.4%

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

      1. expm1-log1p-u 71.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y} \cdot \sinh y\right)\right)} \]

      2. expm1-udef 71.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{y} \cdot \sinh y\right)} - 1} \]

      3. associate-*l/ 71.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \sinh y}{y}}\right)} - 1 \]

      4. *-un-lft-identity 71.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sinh y}}{y}\right)} - 1 \]

   6. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 71.4%

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

   8. Simplified 71.4%

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

   if 1.04999999999999999e148 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. associate-*r/ 3.1%

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

   6. Applied egg-rr 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. *-commutative 3.1%

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

      3. lft-mult-inverse 3.1%

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

      4. associate-*l* 3.1%

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

      5. div-inv 3.1%

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

      6. associate-*l/ 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0036:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \left(\cos x \cdot y\right)}{y}}{y}\\ \end{array} \]

Alternative 3: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0048:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.0048) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0048) {
		tmp = cos(x);
	} else {
		tmp = sinh(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 <= 0.0048d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0048) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0048:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0048)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0048)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0048], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00479999999999999958

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if 0.00479999999999999958 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 80.0%

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

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y} \cdot \sinh y\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{y} \cdot \sinh y\right)} - 1} \]

      3. associate-*l/ 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \sinh y}{y}}\right)} - 1 \]

      4. *-un-lft-identity 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sinh y}}{y}\right)} - 1 \]

   6. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]

      2. expm1-log1p 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0048:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 4: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 500:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + -0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 500.0) (cos x) (* y (+ (/ 1.0 y) (* -0.5 (* x (* x (/ 1.0 y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 500.0) {
		tmp = cos(x);
	} else {
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 500.0d0) then
        tmp = cos(x)
    else
        tmp = y * ((1.0d0 / y) + ((-0.5d0) * (x * (x * (1.0d0 / y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 500.0) {
		tmp = Math.cos(x);
	} else {
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 500.0:
		tmp = math.cos(x)
	else:
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 500.0)
		tmp = cos(x);
	else
		tmp = Float64(y * Float64(Float64(1.0 / y) + Float64(-0.5 * Float64(x * Float64(x * Float64(1.0 / y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 500.0)
		tmp = cos(x);
	else
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 500.0], N[Cos[x], $MachinePrecision], N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(-0.5 * N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 64.8%

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

   if 500 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 13.0%

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

      1. div-inv 13.0%

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

      2. unpow2 13.0%

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

      3. associate-*l* 13.0%

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

   7. Applied egg-rr 13.0%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 500:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + -0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]

Alternative 5: 31.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + -0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-12) 1.0 (* y (+ (/ 1.0 y) (* -0.5 (* x (* x (/ 1.0 y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-12) {
		tmp = 1.0;
	} else {
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-12) then
        tmp = 1.0d0
    else
        tmp = y * ((1.0d0 / y) + ((-0.5d0) * (x * (x * (1.0d0 / y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-12) {
		tmp = 1.0;
	} else {
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-12:
		tmp = 1.0
	else:
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-12)
		tmp = 1.0;
	else
		tmp = Float64(y * Float64(Float64(1.0 / y) + Float64(-0.5 * Float64(x * Float64(x * Float64(1.0 / y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-12)
		tmp = 1.0;
	else
		tmp = y * ((1.0 / y) + (-0.5 * (x * (x * (1.0 / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-12], 1.0, N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(-0.5 * N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-12

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 61.5%

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

   5. Taylor expanded in y around 0 38.5%

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

   6. Taylor expanded in y around 0 38.5%

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

   if 3.2000000000000001e-12 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 4.8%

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

   5. Taylor expanded in x around 0 14.4%

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

      1. div-inv 14.4%

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

      2. unpow2 14.4%

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

      3. associate-*l* 14.4%

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

   7. Applied egg-rr 14.4%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + -0.5 \cdot \left(x \cdot \left(x \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]

Alternative 6: 28.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 99.9%

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

   2. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 66.4%

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

5. Taylor expanded in y around 0 29.7%

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

6. Taylor expanded in y around 0 29.7%

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

7. Final simplification 29.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))