Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• 49.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.9%

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

   2. *-un-lft-identity 78.9%

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

   3. log-prod 78.9%

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

   4. metadata-eval 78.9%

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

   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/ 90.9%

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

   3. associate-*l/ 87.8%

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

   4. associate-/r/ 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.0000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0000002) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0000002) {
		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 <= 1.0000002d0) 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 <= 1.0000002) {
		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 <= 1.0000002:
		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 <= 1.0000002)
		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 <= 1.0000002)
		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, 1.0000002], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.00000019999999989

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.3%

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

   if 1.00000019999999989 < (/.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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 69.9%

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

      1. associate-*r/ 69.1%

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

      2. *-commutative 69.1%

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

      3. associate-/r* 69.1%

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

      4. associate-*r/ 69.9%

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

      5. *-commutative 69.9%

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

      6. associate-/r/ 69.9%

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

      7. rec-exp 69.9%

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

      8. sinh-def 70.0%

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

   7. Simplified 70.0%

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

      1. associate-/r/ 70.0%

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

      2. clear-num 70.0%

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

   9. Applied egg-rr 70.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.0000002:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e+59) (sin x) (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.45e+59) {
		tmp = sin(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 (y <= 1.45d+59) then
        tmp = sin(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 (y <= 1.45e+59) {
		tmp = Math.sin(x);
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.45e+59:
		tmp = math.sin(x)
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.45e+59)
		tmp = sin(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 (y <= 1.45e+59)
		tmp = sin(x);
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.45e+59], N[Sin[x], $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.44999999999999995e59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.5%

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

   if 1.44999999999999995e59 < 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%

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

   4. Applied egg-rr 100.0%

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

   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 19.4%

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

      1. associate-/r/ 19.4%

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

      2. *-commutative 19.4%

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

   10. Applied egg-rr 19.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 310000000:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 310000000.0)
   (/ 1.0 (+ (* x 0.16666666666666666) (/ 1.0 x)))
   (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 310000000.0) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / 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 (y <= 310000000.0d0) then
        tmp = 1.0d0 / ((x * 0.16666666666666666d0) + (1.0d0 / 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 (y <= 310000000.0) {
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 310000000.0:
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x))
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 310000000.0)
		tmp = Float64(1.0 / Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / 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 (y <= 310000000.0)
		tmp = 1.0 / ((x * 0.16666666666666666) + (1.0 / x));
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 310000000.0], N[(1.0 / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1e8

   1. Initial program 100.0%

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

      1. associate-*r/ 87.8%

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

      2. clear-num 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in y around 0 66.7%

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

   6. Taylor expanded in x around 0 31.9%

      \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot x + \frac{1}{x}}} \]

   if 3.1e8 < 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%

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

   4. Applied egg-rr 100.0%

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

   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 16.5%

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

      1. associate-/r/ 16.5%

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

      2. *-commutative 16.5%

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

   10. Applied egg-rr 16.5%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 310000000:\\ \;\;\;\;\frac{1}{x \cdot 0.16666666666666666 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.2e+18) x (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.2e+18) {
		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 <= 2.2d+18) 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 <= 2.2e+18) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.2e+18:
		tmp = x
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.2e+18)
		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 <= 2.2e+18)
		tmp = x;
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.2e+18], x, N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e18

   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%

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

   4. Applied egg-rr 87.3%

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

   5. Taylor expanded in y around 0 50.9%

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

   6. Taylor expanded in x around 0 32.5%

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

   if 2.2e18 < 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 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 48.5%

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

      1. *-commutative 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in x around 0 14.1%

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

      1. associate-/r/ 14.1%

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

      2. *-commutative 14.1%

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

   10. Applied egg-rr 14.1%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.4% 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/ 90.9%

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

   2. clear-num 90.8%

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

4. Applied egg-rr 90.8%

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

5. Taylor expanded in y around 0 50.2%

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

6. Taylor expanded in x around 0 24.0%

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

7. Final simplification 24.0%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))