Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 7

Speedup: 1.0×

• 49.8% 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: 61.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+108} \lor \neg \left(y \leq 7.62 \cdot 10^{+196}\right) \land y \leq 1.4 \cdot 10^{+208}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e+36)
   (sin x)
   (if (or (<= y 9.5e+108) (and (not (<= y 7.62e+196)) (<= y 1.4e+208)))
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (* 0.16666666666666666 (* x (pow y 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+36) {
		tmp = sin(x);
	} else if ((y <= 9.5e+108) || (!(y <= 7.62e+196) && (y <= 1.4e+208))) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = 0.16666666666666666 * (x * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d+36) then
        tmp = sin(x)
    else if ((y <= 9.5d+108) .or. (.not. (y <= 7.62d+196)) .and. (y <= 1.4d+208)) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = 0.16666666666666666d0 * (x * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e+36) {
		tmp = Math.sin(x);
	} else if ((y <= 9.5e+108) || (!(y <= 7.62e+196) && (y <= 1.4e+208))) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = 0.16666666666666666 * (x * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e+36:
		tmp = math.sin(x)
	elif (y <= 9.5e+108) or (not (y <= 7.62e+196) and (y <= 1.4e+208)):
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = 0.16666666666666666 * (x * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e+36)
		tmp = sin(x);
	elseif ((y <= 9.5e+108) || (!(y <= 7.62e+196) && (y <= 1.4e+208)))
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(x * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e+36)
		tmp = sin(x);
	elseif ((y <= 9.5e+108) || (~((y <= 7.62e+196)) && (y <= 1.4e+208)))
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e+36], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 9.5e+108], And[N[Not[LessEqual[y, 7.62e+196]], $MachinePrecision], LessEqual[y, 1.4e+208]]], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.80000000000000025e36

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.3%

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

   if 3.80000000000000025e36 < y < 9.50000000000000097e108 or 7.6199999999999996e196 < y < 1.40000000000000011e208

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.6%

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

   6. Taylor expanded in x around 0 29.8%

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

      1. distribute-rgt-in 29.8%

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

      2. *-lft-identity 29.8%

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

      3. associate-*l* 29.8%

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

      4. unpow2 29.8%

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

      5. unpow3 29.8%

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

   8. Simplified 29.8%

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

   if 9.50000000000000097e108 < y < 7.6199999999999996e196 or 1.40000000000000011e208 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 77.5%

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

   4. Taylor expanded in x around 0 56.1%

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

   5. Taylor expanded in y around inf 56.1%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+108} \lor \neg \left(y \leq 7.62 \cdot 10^{+196}\right) \land y \leq 1.4 \cdot 10^{+208}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.62 \cdot 10^{+196}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (* -0.16666666666666666 (pow x 3.0)))))
   (if (<= y 3.8e+36)
     (sin x)
     (if (<= y 8.6e+108)
       t_0
       (if (<= y 7.62e+196)
         (* x (+ 1.0 (* 0.16666666666666666 (pow y 2.0))))
         (if (<= y 1.4e+208)
           t_0
           (* 0.16666666666666666 (* x (pow y 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = x + (-0.16666666666666666 * pow(x, 3.0));
	double tmp;
	if (y <= 3.8e+36) {
		tmp = sin(x);
	} else if (y <= 8.6e+108) {
		tmp = t_0;
	} else if (y <= 7.62e+196) {
		tmp = x * (1.0 + (0.16666666666666666 * pow(y, 2.0)));
	} else if (y <= 1.4e+208) {
		tmp = t_0;
	} else {
		tmp = 0.16666666666666666 * (x * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    if (y <= 3.8d+36) then
        tmp = sin(x)
    else if (y <= 8.6d+108) then
        tmp = t_0
    else if (y <= 7.62d+196) then
        tmp = x * (1.0d0 + (0.16666666666666666d0 * (y ** 2.0d0)))
    else if (y <= 1.4d+208) then
        tmp = t_0
    else
        tmp = 0.16666666666666666d0 * (x * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	double tmp;
	if (y <= 3.8e+36) {
		tmp = Math.sin(x);
	} else if (y <= 8.6e+108) {
		tmp = t_0;
	} else if (y <= 7.62e+196) {
		tmp = x * (1.0 + (0.16666666666666666 * Math.pow(y, 2.0)));
	} else if (y <= 1.4e+208) {
		tmp = t_0;
	} else {
		tmp = 0.16666666666666666 * (x * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (-0.16666666666666666 * math.pow(x, 3.0))
	tmp = 0
	if y <= 3.8e+36:
		tmp = math.sin(x)
	elif y <= 8.6e+108:
		tmp = t_0
	elif y <= 7.62e+196:
		tmp = x * (1.0 + (0.16666666666666666 * math.pow(y, 2.0)))
	elif y <= 1.4e+208:
		tmp = t_0
	else:
		tmp = 0.16666666666666666 * (x * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)))
	tmp = 0.0
	if (y <= 3.8e+36)
		tmp = sin(x);
	elseif (y <= 8.6e+108)
		tmp = t_0;
	elseif (y <= 7.62e+196)
		tmp = Float64(x * Float64(1.0 + Float64(0.16666666666666666 * (y ^ 2.0))));
	elseif (y <= 1.4e+208)
		tmp = t_0;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (-0.16666666666666666 * (x ^ 3.0));
	tmp = 0.0;
	if (y <= 3.8e+36)
		tmp = sin(x);
	elseif (y <= 8.6e+108)
		tmp = t_0;
	elseif (y <= 7.62e+196)
		tmp = x * (1.0 + (0.16666666666666666 * (y ^ 2.0)));
	elseif (y <= 1.4e+208)
		tmp = t_0;
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.8e+36], N[Sin[x], $MachinePrecision], If[LessEqual[y, 8.6e+108], t$95$0, If[LessEqual[y, 7.62e+196], N[(x * N[(1.0 + N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+208], t$95$0, N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.80000000000000025e36

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.3%

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

   if 3.80000000000000025e36 < y < 8.59999999999999993e108 or 7.6199999999999996e196 < y < 1.40000000000000011e208

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.6%

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

   6. Taylor expanded in x around 0 29.8%

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

      1. distribute-rgt-in 29.8%

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

      2. *-lft-identity 29.8%

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

      3. associate-*l* 29.8%

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

      4. unpow2 29.8%

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

      5. unpow3 29.8%

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

   8. Simplified 29.8%

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

   if 8.59999999999999993e108 < y < 7.6199999999999996e196

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 55.1%

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

   4. Taylor expanded in x around 0 49.0%

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

   if 1.40000000000000011e208 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in x around 0 63.2%

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

   5. Taylor expanded in y around inf 63.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+36}:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+108}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{elif}\;y \leq 7.62 \cdot 10^{+196}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+208}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 425:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 425.0) (sin x) (* 0.16666666666666666 (* x (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 425.0) {
		tmp = sin(x);
	} else {
		tmp = 0.16666666666666666 * (x * pow(y, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 425.0d0) then
        tmp = sin(x)
    else
        tmp = 0.16666666666666666d0 * (x * (y ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 425.0) {
		tmp = Math.sin(x);
	} else {
		tmp = 0.16666666666666666 * (x * Math.pow(y, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 425.0:
		tmp = math.sin(x)
	else:
		tmp = 0.16666666666666666 * (x * math.pow(y, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 425.0)
		tmp = sin(x);
	else
		tmp = Float64(0.16666666666666666 * Float64(x * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 425.0)
		tmp = sin(x);
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 425.0], N[Sin[x], $MachinePrecision], N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 425

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.3%

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

   if 425 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 53.6%

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

   4. Taylor expanded in x around 0 38.5%

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

   5. Taylor expanded in y around inf 38.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 425:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+46}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 7.5e+46) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.5e+46) {
		tmp = sin(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 <= 7.5d+46) then
        tmp = sin(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 <= 7.5e+46) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.5e+46:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.5e+46)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.5e+46)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.5e+46], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000003e46

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.7%

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

   if 7.5000000000000003e46 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

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

   6. Taylor expanded in x around 0 12.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+46}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5e+45) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+45) {
		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 (x <= 5d+45) 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 (x <= 5e+45) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e+45:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+45)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5e+45)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+45], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e45

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.4%

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

   4. Taylor expanded in x around 0 51.9%

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

   5. Taylor expanded in y around 0 32.3%

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

   if 5e45 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 46.2%

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

   6. Taylor expanded in x around 0 19.8%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 77.1%

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

4. Taylor expanded in x around 0 45.8%

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

5. Taylor expanded in y around 0 26.0%

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

6. Final simplification 26.0%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))