Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

Alternatives: 8

Speedup: 1.0×

• 50.0% 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: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 115000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 115000.0) (sin x) (* x (/ (sinh y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 115000.0) {
		tmp = sin(x);
	} else {
		tmp = x * (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 <= 115000.0d0) then
        tmp = sin(x)
    else
        tmp = x * (sinh(y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 115000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 115000.0:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 115000.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 115000.0)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 115000.0], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 115000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.0%

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

   if 115000 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.0%

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

Alternative 3: 55.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 680:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 680.0)
   (sin x)
   (* x (+ 1.0 (* (/ (* x (* x y)) y) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 680.0) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 680.0d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + (((x * (x * y)) / y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 680.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 680.0:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 680.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(Float64(x * Float64(x * y)) / y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 680.0)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 680.0], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 680

   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 680 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.7%

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

   4. Taylor expanded in x around 0 21.5%

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

      1. *-commutative 21.5%

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

   6. Simplified 21.5%

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

      1. unpow2 21.5%

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

   8. Applied egg-rr 21.5%

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

      1. *-un-lft-identity 21.5%

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

      2. associate-*r* 21.5%

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

      3. *-inverses 21.5%

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

      4. associate-/l* 22.9%

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

      5. associate-*l/ 24.4%

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

      6. *-commutative 24.4%

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

   10. Applied egg-rr 24.4%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 680:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.7% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.9e+182)
   (* x (+ 1.0 (* -0.16666666666666666 (* x x))))
   (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.9e+182) {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * 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 <= 2.9d+182) then
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (x * 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 <= 2.9e+182) {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.9e+182:
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)))
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.9e+182)
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.9e+182)
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.9e+182], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e182

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 54.7%

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

   4. Taylor expanded in x around 0 40.7%

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

      1. *-commutative 40.7%

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

   6. Simplified 40.7%

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

      1. unpow2 40.7%

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

   8. Applied egg-rr 40.7%

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

   if 2.8999999999999998e182 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 100.0%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 2.5%

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

   8. Taylor expanded in x around 0 16.9%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+182}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.7% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* (/ (* x (* x y)) y) -0.16666666666666666))))

C

double code(double x, double y) {
	return x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (((x * (x * y)) / y) * (-0.16666666666666666d0)))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
}

Python

def code(x, y):
	return x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(Float64(Float64(x * Float64(x * y)) / y) * -0.16666666666666666)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 49.4%

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

4. Taylor expanded in x around 0 37.6%

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

   1. *-commutative 37.6%

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

6. Simplified 37.6%

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

   1. unpow2 37.6%

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

8. Applied egg-rr 37.6%

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

   1. *-un-lft-identity 37.6%

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

   2. associate-*r* 37.6%

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

   3. *-inverses 37.6%

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

   4. associate-/l* 37.9%

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

   5. associate-*l/ 39.0%

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

   6. *-commutative 39.0%

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

10. Applied egg-rr 39.0%

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

11. Final simplification 39.0%

    \[\leadsto x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right) \]
12. Add Preprocessing

Alternative 6: 29.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.25e+82) x (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+82) {
		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 (x <= 1.25d+82) 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 (x <= 1.25e+82) {
		tmp = x;
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.25e+82:
		tmp = x
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e+82)
		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 (x <= 1.25e+82)
		tmp = x;
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e+82], x, N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000004e82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.3%

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

   4. Taylor expanded in y around 0 35.8%

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

   if 1.25000000000000004e82 < x

   1. Initial program 100.0%

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

      1. add-log-exp 99.7%

         \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 99.7%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 99.7%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 43.6%

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

   8. Taylor expanded in x around 0 22.6%

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

      1. clear-num 22.6%

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

      2. associate-/r/ 22.6%

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

      3. *-commutative 22.6%

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

   10. Applied egg-rr 22.6%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+79}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+79) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+79) {
		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 <= 1d+79) 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 <= 1e+79) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+79:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+79)
		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 <= 1e+79)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+79], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999967e78

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.3%

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

   4. Taylor expanded in y around 0 35.8%

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

   if 9.99999999999999967e78 < x

   1. Initial program 100.0%

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

      1. add-log-exp 99.7%

         \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      2. *-un-lft-identity 99.7%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      3. log-prod 99.7%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 43.6%

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

   8. Taylor expanded in x around 0 22.6%

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

Alternative 8: 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. Taylor expanded in x around 0 68.4%

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

4. Taylor expanded in y around 0 30.3%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))