Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 7

Speedup: 1.0×

• 50.8% 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. Add Preprocessing

Alternative 2: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 2.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 2.0d0) then
        tmp = sin(x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 2.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.4%

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

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

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 72.8%

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

      1. associate-*r/ 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-/r* 72.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2 \cdot y}{e^{y} - \frac{1}{e^{y}}}}{x}}} \]

      4. associate-*r/ 72.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot \frac{y}{e^{y} - \frac{1}{e^{y}}}}}{x}} \]

      5. *-commutative 72.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{e^{y} - \frac{1}{e^{y}}} \cdot 2}}{x}} \]

      6. associate-/r/ 72.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{y}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}}{x}} \]

      7. rec-exp 72.8%

         \[\leadsto \frac{1}{\frac{\frac{y}{\frac{e^{y} - \color{blue}{e^{-y}}}{2}}}{x}} \]

      8. sinh-def 72.8%

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

   7. Simplified 72.8%

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

      1. associate-/r/ 72.8%

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

      2. clear-num 72.8%

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

   9. Applied egg-rr 72.8%

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

4. Final simplification 85.9%

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

Alternative 3: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 105000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 105000.0) (sin x) (* -0.16666666666666666 (pow x 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 105000.0) {
		tmp = sin(x);
	} else {
		tmp = -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 <= 105000.0d0) then
        tmp = sin(x)
    else
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 105000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 105000.0:
		tmp = math.sin(x)
	else:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 105000.0)
		tmp = sin(x);
	else
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 105000.0)
		tmp = sin(x);
	else
		tmp = -0.16666666666666666 * (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 105000.0], N[Sin[x], $MachinePrecision], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 105000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.2%

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

   if 105000 < 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%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x}}} \]

   6. Taylor expanded in x around 0 20.4%

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

      1. distribute-rgt-in 20.4%

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

      2. *-lft-identity 20.4%

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

      3. associate-*l* 20.4%

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

      4. unpow2 20.4%

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

      5. unpow3 20.4%

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

   8. Simplified 20.4%

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

   9. Taylor expanded in x around inf 20.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e+145) (sin x) (/ 1.0 (* y (/ (/ 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+145) {
		tmp = sin(x);
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d+145) then
        tmp = sin(x)
    else
        tmp = 1.0d0 / (y * ((1.0d0 / y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+145) {
		tmp = Math.sin(x);
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e+145:
		tmp = math.sin(x)
	else:
		tmp = 1.0 / (y * ((1.0 / y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e+145)
		tmp = sin(x);
	else
		tmp = Float64(1.0 / Float64(y * Float64(Float64(1.0 / y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e+145)
		tmp = sin(x);
	else
		tmp = 1.0 / (y * ((1.0 / y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e+145], N[Sin[x], $MachinePrecision], N[(1.0 / N[(y * N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.09999999999999989e145

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 56.7%

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

   if 2.09999999999999989e145 < 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%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.8%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \sin x}}} \]
   6. Step-by-step derivation

      1. *-commutative 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in x around 0 30.4%

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

      1. clear-num 30.4%

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

      2. associate-/r/ 30.4%

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

      3. *-commutative 30.4%

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

      4. associate-/r* 30.4%

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

   10. Applied egg-rr 30.4%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e+93) x (/ 1.0 (* y (/ (/ 1.0 y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e+93) {
		tmp = x;
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / x));
	}
	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+93) then
        tmp = x
    else
        tmp = 1.0d0 / (y * ((1.0d0 / y) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e+93) {
		tmp = x;
	} else {
		tmp = 1.0 / (y * ((1.0 / y) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e+93:
		tmp = x
	else:
		tmp = 1.0 / (y * ((1.0 / y) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e+93)
		tmp = x;
	else
		tmp = Float64(1.0 / Float64(y * Float64(Float64(1.0 / y) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e+93)
		tmp = x;
	else
		tmp = 1.0 / (y * ((1.0 / y) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e+93], x, N[(1.0 / N[(y * N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000019e93

   1. Initial program 100.0%

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

      1. associate-*r/ 85.7%

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

      2. clear-num 85.6%

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

      3. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in y around 0 57.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x}}} \]

   6. Taylor expanded in x around 0 29.2%

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

   if 3.10000000000000019e93 < 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%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.7%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \sin x}}} \]
   6. Step-by-step derivation

      1. *-commutative 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in x around 0 25.5%

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

      1. clear-num 25.5%

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

      2. associate-/r/ 25.5%

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

      3. *-commutative 25.5%

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

      4. associate-/r* 25.5%

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

   10. Applied egg-rr 25.5%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{\frac{1}{y}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 20000000000000.0) x (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 20000000000000.0) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 20000000000000.0d0) then
        tmp = x
    else
        tmp = (1.0d0 / y) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 20000000000000.0) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 20000000000000.0:
		tmp = x
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 20000000000000.0)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 20000000000000.0)
		tmp = x;
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 20000000000000.0], x, N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e13

   1. Initial program 100.0%

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

      1. associate-*r/ 83.6%

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

      2. clear-num 83.5%

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

      3. *-commutative 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in y around 0 53.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x}}} \]

   6. Taylor expanded in x around 0 33.3%

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

   if 2e13 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 46.2%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \sin x}}} \]
   6. Step-by-step derivation

      1. *-commutative 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in x around 0 8.8%

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

      1. associate-/r/ 8.8%

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

      2. *-commutative 8.8%

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

   10. Applied egg-rr 8.8%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 20000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.7% 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/ 87.4%

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

   2. clear-num 87.3%

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

   3. *-commutative 87.3%

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

4. Applied egg-rr 87.3%

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

5. Taylor expanded in y around 0 51.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x}}} \]

6. Taylor expanded in x around 0 26.2%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))