Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 8

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]

2. Final simplification 100.0%

   \[\leadsto \cos x \cdot \frac{\sinh y}{y} \]

Alternative 2: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{2}{\sinh y}}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sinh y) y) 1.0) (cos x) (/ (/ 2.0 (/ 2.0 (sinh y))) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0) {
		tmp = cos(x);
	} else {
		tmp = (2.0 / (2.0 / 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) <= 1.0d0) then
        tmp = cos(x)
    else
        tmp = (2.0d0 / (2.0d0 / 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) <= 1.0) {
		tmp = Math.cos(x);
	} else {
		tmp = (2.0 / (2.0 / Math.sinh(y))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sinh(y) / y) <= 1.0:
		tmp = math.cos(x)
	else:
		tmp = (2.0 / (2.0 / math.sinh(y))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.0)
		tmp = cos(x);
	else
		tmp = Float64(Float64(2.0 / Float64(2.0 / sinh(y))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) / y) <= 1.0)
		tmp = cos(x);
	else
		tmp = (2.0 / (2.0 / sinh(y))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], 1.0], N[Cos[x], $MachinePrecision], N[(N[(2.0 / N[(2.0 / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 73.9%

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

   5. Taylor expanded in y around inf 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]

   7. Simplified 73.9%

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

      1. sinh-def 73.9%

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

      2. sinh-undef 73.9%

         \[\leadsto \frac{\frac{\color{blue}{2 \cdot \sinh y}}{2}}{y} \]

      3. associate-/l* 73.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{2}{\sinh y}}}}{y} \]

   9. Applied egg-rr 73.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{2}{\sinh y}}}{y}\\ \end{array} \]

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (cos x) t_0)))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		tmp = cos(x);
	} else {
		tmp = 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 <= 1.0d0) then
        tmp = cos(x)
    else
        tmp = 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 <= 1.0) {
		tmp = Math.cos(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.0:
		tmp = math.cos(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = cos(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.0)
		tmp = cos(x);
	else
		tmp = 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, 1.0], N[Cos[x], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 73.9%

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

   5. Taylor expanded in y around inf 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{y} - \frac{1}{e^{y}}}{y}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]

   7. Simplified 73.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 4: 61.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x \cdot x}\\ \mathbf{if}\;y \leq 4500:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* x x))))
   (if (<= y 4500.0)
     (cos x)
     (if (<= y 4.7e+70)
       t_0
       (if (<= y 2e+92)
         (+ 1.0 (* (* x x) -0.5))
         (if (<= y 2.45e+147) t_0 (+ 1.0 (* 0.16666666666666666 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (x * x);
	double tmp;
	if (y <= 4500.0) {
		tmp = cos(x);
	} else if (y <= 4.7e+70) {
		tmp = t_0;
	} else if (y <= 2e+92) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else if (y <= 2.45e+147) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	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 = 2.0d0 / (x * x)
    if (y <= 4500.0d0) then
        tmp = cos(x)
    else if (y <= 4.7d+70) then
        tmp = t_0
    else if (y <= 2d+92) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else if (y <= 2.45d+147) then
        tmp = t_0
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 / (x * x);
	double tmp;
	if (y <= 4500.0) {
		tmp = Math.cos(x);
	} else if (y <= 4.7e+70) {
		tmp = t_0;
	} else if (y <= 2e+92) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else if (y <= 2.45e+147) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (x * x)
	tmp = 0
	if y <= 4500.0:
		tmp = math.cos(x)
	elif y <= 4.7e+70:
		tmp = t_0
	elif y <= 2e+92:
		tmp = 1.0 + ((x * x) * -0.5)
	elif y <= 2.45e+147:
		tmp = t_0
	else:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(x * x))
	tmp = 0.0
	if (y <= 4500.0)
		tmp = cos(x);
	elseif (y <= 4.7e+70)
		tmp = t_0;
	elseif (y <= 2e+92)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	elseif (y <= 2.45e+147)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 / (x * x);
	tmp = 0.0;
	if (y <= 4500.0)
		tmp = cos(x);
	elseif (y <= 4.7e+70)
		tmp = t_0;
	elseif (y <= 2e+92)
		tmp = 1.0 + ((x * x) * -0.5);
	elseif (y <= 2.45e+147)
		tmp = t_0;
	else
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4500.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 4.7e+70], t$95$0, If[LessEqual[y, 2e+92], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e+147], t$95$0, N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4500

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 67.4%

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

   if 4500 < y < 4.6999999999999998e70 or 2.0000000000000001e92 < y < 2.4499999999999999e147

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. clear-num 3.1%

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

      3. un-div-inv 3.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   7. Taylor expanded in x around 0 2.6%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left({x}^{2} \cdot y\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 2.6%

         \[\leadsto \frac{y}{y + 0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}} \]

      2. unpow2 2.6%

         \[\leadsto \frac{y}{y + 0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   9. Simplified 2.6%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}} \]

   10. Taylor expanded in x around inf 23.6%

       \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 23.6%

          \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]

   12. Simplified 23.6%

       \[\leadsto \color{blue}{\frac{2}{x \cdot x}} \]

   if 4.6999999999999998e70 < y < 2.0000000000000001e92

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 51.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 51.0%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Simplified 51.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]

   if 2.4499999999999999e147 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 75.8%

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

   5. Taylor expanded in y around 0 73.1%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 73.1%

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

   7. Simplified 73.1%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4500:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+70}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5: 33.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4500:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+70} \lor \neg \left(y \leq 6.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4500.0)
   1.0
   (if (or (<= y 2.15e+70) (not (<= y 6.5e+95)))
     (/ 2.0 (* x x))
     (+ 1.0 (* (* x x) -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4500.0) {
		tmp = 1.0;
	} else if ((y <= 2.15e+70) || !(y <= 6.5e+95)) {
		tmp = 2.0 / (x * x);
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4500.0d0) then
        tmp = 1.0d0
    else if ((y <= 2.15d+70) .or. (.not. (y <= 6.5d+95))) then
        tmp = 2.0d0 / (x * x)
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4500.0) {
		tmp = 1.0;
	} else if ((y <= 2.15e+70) || !(y <= 6.5e+95)) {
		tmp = 2.0 / (x * x);
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4500.0:
		tmp = 1.0
	elif (y <= 2.15e+70) or not (y <= 6.5e+95):
		tmp = 2.0 / (x * x)
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4500.0)
		tmp = 1.0;
	elseif ((y <= 2.15e+70) || !(y <= 6.5e+95))
		tmp = Float64(2.0 / Float64(x * x));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4500.0)
		tmp = 1.0;
	elseif ((y <= 2.15e+70) || ~((y <= 6.5e+95)))
		tmp = 2.0 / (x * x);
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4500.0], 1.0, If[Or[LessEqual[y, 2.15e+70], N[Not[LessEqual[y, 6.5e+95]], $MachinePrecision]], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4500

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.6%

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

   5. Taylor expanded in y around 0 38.3%

      \[\leadsto \color{blue}{1} \]

   if 4500 < y < 2.15e70 or 6.5e95 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. clear-num 3.1%

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

      3. un-div-inv 3.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   7. Taylor expanded in x around 0 2.3%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left({x}^{2} \cdot y\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{y}{y + 0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}} \]

      2. unpow2 2.3%

         \[\leadsto \frac{y}{y + 0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   9. Simplified 2.3%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}} \]

   10. Taylor expanded in x around inf 16.1%

       \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 16.1%

          \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]

   12. Simplified 16.1%

       \[\leadsto \color{blue}{\frac{2}{x \cdot x}} \]

   if 2.15e70 < y < 6.5e95

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 51.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 51.0%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Simplified 51.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4500:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+70} \lor \neg \left(y \leq 6.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 6: 48.1% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+164}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.9e+164)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.9e+164) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.9d+164) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.9e+164) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.9e+164:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.9e+164)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.9e+164)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.9e+164], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999985e164

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.8%

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

   5. Taylor expanded in y around 0 50.8%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 50.8%

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

   7. Simplified 50.8%

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

   if 3.89999999999999985e164 < x

   1. Initial program 99.9%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 43.9%

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

   5. Taylor expanded in x around 0 35.1%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 35.1%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Simplified 35.1%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+164}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 7: 34.0% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4500.0) 1.0 (/ 2.0 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4500.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4500.0d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4500.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4500.0:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4500.0)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4500.0)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4500.0], 1.0, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4500

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.6%

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

   5. Taylor expanded in y around 0 38.3%

      \[\leadsto \color{blue}{1} \]

   if 4500 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

      1. *-commutative 3.1%

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

      2. clear-num 3.1%

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

      3. un-div-inv 3.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   6. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y}{\frac{y}{\cos x}}} \]

   7. Taylor expanded in x around 0 2.3%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left({x}^{2} \cdot y\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{y}{y + 0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}} \]

      2. unpow2 2.3%

         \[\leadsto \frac{y}{y + 0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   9. Simplified 2.3%

      \[\leadsto \frac{y}{\color{blue}{y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}} \]

   10. Taylor expanded in x around inf 13.9%

       \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 13.9%

          \[\leadsto \frac{2}{\color{blue}{x \cdot x}} \]

   12. Simplified 13.9%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]

Alternative 8: 28.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Step-by-step derivation

   1. associate-*r/ 100.0%

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

   2. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 65.0%

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

5. Taylor expanded in y around 0 29.6%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 29.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))