Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 7

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. Final simplification 100.0%

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

Alternative 2: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-10) (sin x) (* x (+ 1.0 (* 0.16666666666666666 (pow y 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-10) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * 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 <= 1.7d-10) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + (0.16666666666666666d0 * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-10) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.7e-10:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (0.16666666666666666 * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.7e-10)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(0.16666666666666666 * (y ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.7e-10)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (0.16666666666666666 * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.7e-10], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000007e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 73.4%

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

   if 1.70000000000000007e-10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 48.7%

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

      1. associate-*r* 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around 0 49.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-10) (sin x) (+ x (* 0.16666666666666666 (* x (pow y 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-10) {
		tmp = sin(x);
	} else {
		tmp = x + (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 <= 1.7d-10) then
        tmp = sin(x)
    else
        tmp = x + (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 <= 1.7e-10) {
		tmp = Math.sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (x * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.7e-10:
		tmp = math.sin(x)
	else:
		tmp = x + (0.16666666666666666 * (x * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.7e-10)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * (y ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.7e-10)
		tmp = sin(x);
	else
		tmp = x + (0.16666666666666666 * (x * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.7e-10], N[Sin[x], $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.70000000000000007e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 73.4%

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

   if 1.70000000000000007e-10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 48.7%

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

      1. associate-*r* 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around 0 49.8%

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

   7. Taylor expanded in y around 0 49.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 700000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 700000.0) (sin x) (* 0.16666666666666666 (* x (pow y 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 700000.0) {
		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 <= 700000.0d0) 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 <= 700000.0) {
		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 <= 700000.0:
		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 <= 700000.0)
		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 <= 700000.0)
		tmp = sin(x);
	else
		tmp = 0.16666666666666666 * (x * (y ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 700000.0], 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 < 7e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.5%

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

   if 7e5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 49.3%

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

      1. associate-*r* 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around 0 50.4%

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

   7. Taylor expanded in y around inf 50.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 700000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot {y}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 235000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 235000.0) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 235000.0) {
		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 <= 235000.0d0) 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 <= 235000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 235000.0:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 235000.0)
		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 <= 235000.0)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 235000.0], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 235000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.5%

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

   if 235000 < 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 2.7%

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

   8. Taylor expanded in x around 0 18.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 235000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.5% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4e+86) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+86) {
		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 <= 4d+86) 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 <= 4e+86) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+86:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+86)
		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 <= 4e+86)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+86], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e86

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

      1. associate-*r* 78.7%

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

   5. Simplified 78.7%

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

   6. Taylor expanded in x around 0 47.7%

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

   7. Taylor expanded in y around 0 35.9%

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

   if 4.0000000000000001e86 < 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 2.8%

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

   8. Taylor expanded in x around 0 27.3%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 y around 0 78.2%

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

   1. associate-*r* 78.2%

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

5. Simplified 78.2%

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

6. Taylor expanded in x around 0 51.4%

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

7. Taylor expanded in y around 0 31.5%

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

8. Final simplification 31.5%

   \[\leadsto x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))