Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 520:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+62} \lor \neg \left(y \leq 4.1 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 520.0)
   (sin x)
   (if (or (<= y 7e+62) (not (<= y 4.1e+95)))
     (log1p (expm1 x))
     (+ x (* -0.16666666666666666 (pow x 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 520.0) {
		tmp = sin(x);
	} else if ((y <= 7e+62) || !(y <= 4.1e+95)) {
		tmp = log1p(expm1(x));
	} else {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 520.0) {
		tmp = Math.sin(x);
	} else if ((y <= 7e+62) || !(y <= 4.1e+95)) {
		tmp = Math.log1p(Math.expm1(x));
	} else {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 520.0:
		tmp = math.sin(x)
	elif (y <= 7e+62) or not (y <= 4.1e+95):
		tmp = math.log1p(math.expm1(x))
	else:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 520.0)
		tmp = sin(x);
	elseif ((y <= 7e+62) || !(y <= 4.1e+95))
		tmp = log1p(expm1(x));
	else
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 520.0], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 7e+62], N[Not[LessEqual[y, 4.1e+95]], $MachinePrecision]], N[Log[1 + N[(Exp[x] - 1), $MachinePrecision]], $MachinePrecision], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 520

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 70.4%

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

   if 520 < y < 6.99999999999999967e62 or 4.09999999999999986e95 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 15.1%

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

      1. div-inv 15.1%

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

      2. associate-*l* 2.8%

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

      3. div-inv 2.8%

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

      4. *-inverses 2.8%

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

      5. *-commutative 2.8%

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

      6. *-un-lft-identity 2.8%

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

      7. log1p-expm1-u 31.4%

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

   8. Applied egg-rr 31.4%

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

   if 6.99999999999999967e62 < y < 4.09999999999999986e95

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.8%

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

   6. Taylor expanded in x around 0 51.2%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 520:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+62} \lor \neg \left(y \leq 4.1 \cdot 10^{+95}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(y \cdot {x}^{3}\right)}{y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+296}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e+61)
   (sin x)
   (if (<= y 3.5e+169)
     (/ (* -0.16666666666666666 (* y (pow x 3.0))) y)
     (if (<= y 1.5e+296)
       (/ (* x y) y)
       (+ x (* -0.16666666666666666 (pow x 3.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.8e+61) {
		tmp = sin(x);
	} else if (y <= 3.5e+169) {
		tmp = (-0.16666666666666666 * (y * pow(x, 3.0))) / y;
	} else if (y <= 1.5e+296) {
		tmp = (x * y) / y;
	} else {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.8d+61) then
        tmp = sin(x)
    else if (y <= 3.5d+169) then
        tmp = ((-0.16666666666666666d0) * (y * (x ** 3.0d0))) / y
    else if (y <= 1.5d+296) then
        tmp = (x * y) / y
    else
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e+61) {
		tmp = Math.sin(x);
	} else if (y <= 3.5e+169) {
		tmp = (-0.16666666666666666 * (y * Math.pow(x, 3.0))) / y;
	} else if (y <= 1.5e+296) {
		tmp = (x * y) / y;
	} else {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.8e+61:
		tmp = math.sin(x)
	elif y <= 3.5e+169:
		tmp = (-0.16666666666666666 * (y * math.pow(x, 3.0))) / y
	elif y <= 1.5e+296:
		tmp = (x * y) / y
	else:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.8e+61)
		tmp = sin(x);
	elseif (y <= 3.5e+169)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * (x ^ 3.0))) / y);
	elseif (y <= 1.5e+296)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e+61)
		tmp = sin(x);
	elseif (y <= 3.5e+169)
		tmp = (-0.16666666666666666 * (y * (x ^ 3.0))) / y;
	elseif (y <= 1.5e+296)
		tmp = (x * y) / y;
	else
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.8e+61], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.5e+169], N[(N[(-0.16666666666666666 * N[(y * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.5e+296], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.8000000000000001e61

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.8%

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

   if 2.8000000000000001e61 < y < 3.50000000000000019e169

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.9%

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

   6. Taylor expanded in x around 0 20.8%

      \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot \left({x}^{3} \cdot y\right) + x \cdot y}}{y} \]

   7. Taylor expanded in x around inf 40.5%

      \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot \left({x}^{3} \cdot y\right)}}{y} \]

   if 3.50000000000000019e169 < y < 1.50000000000000006e296

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 17.3%

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

   if 1.50000000000000006e296 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.5%

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

   6. Taylor expanded in x around 0 2.5%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(y \cdot {x}^{3}\right)}{y}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+296}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+169}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e+61)
   (sin x)
   (if (<= y 2.5e+169)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (/ (* x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+61) {
		tmp = sin(x);
	} else if (y <= 2.5e+169) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = (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 <= 2.65d+61) then
        tmp = sin(x)
    else if (y <= 2.5d+169) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+61) {
		tmp = Math.sin(x);
	} else if (y <= 2.5e+169) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.65e+61:
		tmp = math.sin(x)
	elif y <= 2.5e+169:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e+61)
		tmp = sin(x);
	elseif (y <= 2.5e+169)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e+61)
		tmp = sin(x);
	elseif (y <= 2.5e+169)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e+61], N[Sin[x], $MachinePrecision], If[LessEqual[y, 2.5e+169], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65000000000000026e61

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.8%

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

   if 2.65000000000000026e61 < y < 2.50000000000000009e169

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.9%

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

   6. Taylor expanded in x around 0 28.2%

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

   if 2.50000000000000009e169 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 16.9%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+169}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+129}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e+61)
   (sin x)
   (if (<= y 8e+129) (* -0.16666666666666666 (pow x 3.0)) (/ (* x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+61) {
		tmp = sin(x);
	} else if (y <= 8e+129) {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	} else {
		tmp = (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 <= 2.65d+61) then
        tmp = sin(x)
    else if (y <= 8d+129) then
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+61) {
		tmp = Math.sin(x);
	} else if (y <= 8e+129) {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.65e+61:
		tmp = math.sin(x)
	elif y <= 8e+129:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e+61)
		tmp = sin(x);
	elseif (y <= 8e+129)
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e+61)
		tmp = sin(x);
	elseif (y <= 8e+129)
		tmp = -0.16666666666666666 * (x ^ 3.0);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e+61], N[Sin[x], $MachinePrecision], If[LessEqual[y, 8e+129], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65000000000000026e61

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.8%

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

   if 2.65000000000000026e61 < y < 8e129

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.9%

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

   6. Taylor expanded in x around 0 19.1%

      \[\leadsto \frac{\color{blue}{-0.16666666666666666 \cdot \left({x}^{3} \cdot y\right) + x \cdot y}}{y} \]

   7. Taylor expanded in x around inf 37.3%

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

   if 8e129 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 17.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+61}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+129}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e+15) (sin x) (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+15) {
		tmp = sin(x);
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.3d+15) then
        tmp = sin(x)
    else
        tmp = (x * y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+15) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e+15:
		tmp = math.sin(x)
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e+15)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e+15)
		tmp = sin(x);
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e+15], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.4%

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

   if 4.3e15 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 14.3%

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

      1. clear-num 14.3%

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

      2. associate-/r/ 14.3%

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

   8. Applied egg-rr 14.3%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.1e-12) x (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-12) {
		tmp = x;
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-12) then
        tmp = x
    else
        tmp = (x * y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-12) {
		tmp = x;
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-12:
		tmp = x
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-12)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-12)
		tmp = x;
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-12], x, N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1000000000000001e-12

   1. Initial program 100.0%

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

      1. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in y around 0 56.3%

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

   6. Taylor expanded in x around 0 34.4%

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

   if 3.1000000000000001e-12 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 6.1%

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

   6. Taylor expanded in x around 0 15.0%

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

      1. clear-num 15.0%

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

      2. associate-/r/ 15.0%

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

   8. Applied egg-rr 15.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-12}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.3% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-21) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-21) {
		tmp = x;
	} else {
		tmp = (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 <= 2d-21) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-21) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-21:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-21)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-21)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-21], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 100.0%

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

      1. associate-*r/ 85.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in y around 0 56.3%

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

   6. Taylor expanded in x around 0 34.4%

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

   if 1.99999999999999982e-21 < y

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 6.1%

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

   6. Taylor expanded in x around 0 15.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 89.1%

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

4. Applied egg-rr 89.1%

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

5. Taylor expanded in y around 0 43.9%

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

6. Taylor expanded in x around 0 27.0%

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

7. Final simplification 27.0%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))