Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 10

Speedup: 1.0×

• 49.6% 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 70.1%

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

   2. *-un-lft-identity 70.1%

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

   3. log-prod 70.1%

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

   4. metadata-eval 70.1%

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

   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/ 86.9%

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

   3. associate-*l/ 89.8%

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

   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. Add Preprocessing

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.01:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sinh y) y) 1.01) (sin x) (/ (* x (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.01) {
		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 ((sinh(y) / y) <= 1.01d0) 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 ((Math.sinh(y) / y) <= 1.01) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) / y) <= 1.01:
		tmp = math.sin(x)
	else:
		tmp = (x * math.sinh(y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.01)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) / y) <= 1.01)
		tmp = sin(x);
	else
		tmp = (x * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], 1.01], N[Sin[x], $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.1%

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

   if 1.01000000000000001 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. associate-*r/ 75.8%

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

   5. Applied egg-rr 75.8%

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

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1.01:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.01) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.01) {
		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 <= 1.01d0) 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 <= 1.01) {
		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 <= 1.01:
		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 <= 1.01)
		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 <= 1.01)
		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, 1.01], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.1%

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

   if 1.01000000000000001 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

Alternative 4: 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 5: 54.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 500000:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+69} \lor \neg \left(y \leq 2.3 \cdot 10^{+241}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 500000.0)
   (sin x)
   (if (or (<= y 4.5e+69) (not (<= y 2.3e+241)))
     (* -0.16666666666666666 (pow x 3.0))
     (* (/ 1.0 y) (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 500000.0) {
		tmp = sin(x);
	} else if ((y <= 4.5e+69) || !(y <= 2.3e+241)) {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	} 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 <= 500000.0d0) then
        tmp = sin(x)
    else if ((y <= 4.5d+69) .or. (.not. (y <= 2.3d+241))) then
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    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 <= 500000.0) {
		tmp = Math.sin(x);
	} else if ((y <= 4.5e+69) || !(y <= 2.3e+241)) {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 500000.0:
		tmp = math.sin(x)
	elif (y <= 4.5e+69) or not (y <= 2.3e+241):
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 500000.0)
		tmp = sin(x);
	elseif ((y <= 4.5e+69) || !(y <= 2.3e+241))
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	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 <= 500000.0)
		tmp = sin(x);
	elseif ((y <= 4.5e+69) || ~((y <= 2.3e+241)))
		tmp = -0.16666666666666666 * (x ^ 3.0);
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 500000.0], N[Sin[x], $MachinePrecision], If[Or[LessEqual[y, 4.5e+69], N[Not[LessEqual[y, 2.3e+241]], $MachinePrecision]], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 71.2%

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

   if 5e5 < y < 4.4999999999999999e69 or 2.2999999999999999e241 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.6%

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

   4. Taylor expanded in x around 0 32.4%

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

      1. distribute-rgt-in 32.4%

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

      2. *-lft-identity 32.4%

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

      3. associate-*l* 32.4%

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

      4. pow-plus 32.4%

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

      5. metadata-eval 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in x around inf 32.2%

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

   if 4.4999999999999999e69 < y < 2.2999999999999999e241

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.8%

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

      1. associate-*r/ 82.8%

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

   5. Applied egg-rr 82.8%

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

   6. Taylor expanded in y around 0 15.6%

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

      1. clear-num 15.6%

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

      2. associate-/r/ 15.6%

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

   8. Applied egg-rr 15.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 500000:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+69} \lor \neg \left(y \leq 2.3 \cdot 10^{+241}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+69}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+69) (sin x) (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+69) {
		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 <= 5.5d+69) 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 <= 5.5e+69) {
		tmp = Math.sin(x);
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.5e+69:
		tmp = math.sin(x)
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.5e+69)
		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 <= 5.5e+69)
		tmp = sin(x);
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.5e+69], 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 < 5.50000000000000002e69

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.6%

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

   if 5.50000000000000002e69 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.7%

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

      1. associate-*r/ 78.7%

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

   5. Applied egg-rr 78.7%

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

   6. Taylor expanded in y around 0 16.5%

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

      1. clear-num 16.5%

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

      2. associate-/r/ 16.5%

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

   8. Applied egg-rr 16.5%

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

Alternative 7: 29.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+73}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x \cdot y\right) + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e+73) x (/ (+ (+ 1.0 (* x y)) -1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+73) {
		tmp = x;
	} else {
		tmp = ((1.0 + (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 <= 2.7d+73) then
        tmp = x
    else
        tmp = ((1.0d0 + (x * y)) + (-1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.7e+73) {
		tmp = x;
	} else {
		tmp = ((1.0 + (x * y)) + -1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.7e+73:
		tmp = x
	else:
		tmp = ((1.0 + (x * y)) + -1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e+73)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * y)) + -1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.7e+73)
		tmp = x;
	else
		tmp = ((1.0 + (x * y)) + -1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e+73], x, N[(N[(N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6999999999999999e73

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.9%

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

   4. Taylor expanded in y around 0 39.5%

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

   if 2.6999999999999999e73 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 26.8%

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

      1. associate-*r/ 26.8%

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

   5. Applied egg-rr 26.8%

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

   6. Taylor expanded in y around 0 13.6%

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

      1. expm1-log1p-u 6.1%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y\right)\right)}}{y} \]

      2. expm1-undefine 6.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(x \cdot y\right)} - 1}}{y} \]

      3. log1p-undefine 6.2%

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

      4. rem-exp-log 13.6%

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

   8. Applied egg-rr 13.6%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+73}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x \cdot y\right) + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+44) x (* (/ 1.0 y) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+44) {
		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 <= 1d+44) 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 <= 1e+44) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+44:
		tmp = x
	else:
		tmp = (1.0 / y) * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+44)
		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 <= 1e+44)
		tmp = x;
	else
		tmp = (1.0 / y) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+44], x, N[(N[(1.0 / y), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0000000000000001e44

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.6%

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

   4. Taylor expanded in y around 0 40.6%

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

   if 1.0000000000000001e44 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 25.8%

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

      1. associate-*r/ 25.8%

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

   5. Applied egg-rr 25.8%

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

   6. Taylor expanded in y around 0 12.2%

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

      1. clear-num 12.2%

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

      2. associate-/r/ 12.2%

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

   8. Applied egg-rr 12.2%

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

Alternative 9: 29.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2e+41) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2e+41) {
		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 <= 2d+41) 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 <= 2e+41) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2e+41:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2e+41)
		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 <= 2e+41)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2e+41], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000001e41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.6%

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

   4. Taylor expanded in y around 0 40.6%

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

   if 2.00000000000000001e41 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 25.8%

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

      1. associate-*r/ 25.8%

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

   5. Applied egg-rr 25.8%

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

   6. Taylor expanded in y around 0 12.2%

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

Alternative 10: 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. Taylor expanded in x around 0 67.7%

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

4. Taylor expanded in y around 0 33.3%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))