Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 13.8s

Alternatives: 12

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (/ (sin y) y) (cosh x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Final simplification 99.9%

   \[\leadsto \frac{\sin y}{y} \cdot \cosh x \]

Alternative 2: 87.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+71}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+150}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin y) (/ (+ 1.0 (* 0.5 (* x x))) y))))
   (if (<= x 1.9)
     t_0
     (if (<= x 1e+71)
       (cosh x)
       (if (<= x 1.32e+150)
         (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) * ((1.0 + (0.5 * (x * x))) / y);
	double tmp;
	if (x <= 1.9) {
		tmp = t_0;
	} else if (x <= 1e+71) {
		tmp = cosh(x);
	} else if (x <= 1.32e+150) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} 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 = sin(y) * ((1.0d0 + (0.5d0 * (x * x))) / y)
    if (x <= 1.9d0) then
        tmp = t_0
    else if (x <= 1d+71) then
        tmp = cosh(x)
    else if (x <= 1.32d+150) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) * ((1.0 + (0.5 * (x * x))) / y);
	double tmp;
	if (x <= 1.9) {
		tmp = t_0;
	} else if (x <= 1e+71) {
		tmp = Math.cosh(x);
	} else if (x <= 1.32e+150) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) * ((1.0 + (0.5 * (x * x))) / y)
	tmp = 0
	if x <= 1.9:
		tmp = t_0
	elif x <= 1e+71:
		tmp = math.cosh(x)
	elif x <= 1.32e+150:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / y))
	tmp = 0.0
	if (x <= 1.9)
		tmp = t_0;
	elseif (x <= 1e+71)
		tmp = cosh(x);
	elseif (x <= 1.32e+150)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) * ((1.0 + (0.5 * (x * x))) / y);
	tmp = 0.0;
	if (x <= 1.9)
		tmp = t_0;
	elseif (x <= 1e+71)
		tmp = cosh(x);
	elseif (x <= 1.32e+150)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9], t$95$0, If[LessEqual[x, 1e+71], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.32e+150], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.8999999999999999 or 1.32e150 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 85.6%

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

      1. unpow2 50.9%

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

   8. Simplified 85.6%

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

   if 1.8999999999999999 < x < 1e71

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   if 1e71 < x < 1.32e150

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 70.6%

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

      1. unpow2 8.0%

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

   4. Simplified 70.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 10^{+71}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+150}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 3: 86.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{\sin y}{\frac{y}{t_0}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+150}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (* x x)))))
   (if (<= x 1.9)
     (/ (sin y) (/ y t_0))
     (if (<= x 4e+68)
       (cosh x)
       (if (<= x 1.32e+150)
         (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
         (* (sin y) (/ t_0 y)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 1.9) {
		tmp = sin(y) / (y / t_0);
	} else if (x <= 4e+68) {
		tmp = cosh(x);
	} else if (x <= 1.32e+150) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = sin(y) * (t_0 / 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 = 1.0d0 + (0.5d0 * (x * x))
    if (x <= 1.9d0) then
        tmp = sin(y) / (y / t_0)
    else if (x <= 4d+68) then
        tmp = cosh(x)
    else if (x <= 1.32d+150) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = sin(y) * (t_0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if (x <= 1.9) {
		tmp = Math.sin(y) / (y / t_0);
	} else if (x <= 4e+68) {
		tmp = Math.cosh(x);
	} else if (x <= 1.32e+150) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * (t_0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.5 * (x * x))
	tmp = 0
	if x <= 1.9:
		tmp = math.sin(y) / (y / t_0)
	elif x <= 4e+68:
		tmp = math.cosh(x)
	elif x <= 1.32e+150:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * (t_0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(sin(y) / Float64(y / t_0));
	elseif (x <= 4e+68)
		tmp = cosh(x);
	elseif (x <= 1.32e+150)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sin(y) * Float64(t_0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.5 * (x * x));
	tmp = 0.0;
	if (x <= 1.9)
		tmp = sin(y) / (y / t_0);
	elseif (x <= 4e+68)
		tmp = cosh(x);
	elseif (x <= 1.32e+150)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * (t_0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9], N[(N[Sin[y], $MachinePrecision] / N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e+68], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.32e+150], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 82.8%

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

      1. unpow2 47.6%

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

   6. Simplified 82.8%

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

   if 1.8999999999999999 < x < 3.99999999999999981e68

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   if 3.99999999999999981e68 < x < 1.32e150

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 70.6%

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

      1. unpow2 8.0%

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

   4. Simplified 70.6%

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

   if 1.32e150 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 97.5%

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

      1. unpow2 67.0%

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

   8. Simplified 97.5%

      \[\leadsto \frac{\color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)}}{y} \cdot \sin y \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{\sin y}{\frac{y}{1 + 0.5 \cdot \left(x \cdot x\right)}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+150}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 4: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00046:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 10^{+69}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00046)
   (/ (sin y) y)
   (if (<= x 1e+69)
     (cosh x)
     (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00046) {
		tmp = sin(y) / y;
	} else if (x <= 1e+69) {
		tmp = cosh(x);
	} else {
		tmp = cosh(x) * (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) :: tmp
    if (x <= 0.00046d0) then
        tmp = sin(y) / y
    else if (x <= 1d+69) then
        tmp = cosh(x)
    else
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00046) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1e+69) {
		tmp = Math.cosh(x);
	} else {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00046:
		tmp = math.sin(y) / y
	elif x <= 1e+69:
		tmp = math.cosh(x)
	else:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00046)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1e+69)
		tmp = cosh(x);
	else
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00046)
		tmp = sin(y) / y;
	elseif (x <= 1e+69)
		tmp = cosh(x);
	else
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00046], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1e+69], N[Cosh[x], $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.6000000000000001e-4

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 67.6%

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

   if 4.6000000000000001e-4 < x < 1.0000000000000001e69

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 92.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   if 1.0000000000000001e69 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 78.6%

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

      1. unpow2 14.8%

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

   4. Simplified 78.6%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00046:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 10^{+69}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00031)
   (/ (sin y) y)
   (if (<= x 4e+153)
     (cosh x)
     (* (+ 1.0 (* 0.5 (* x x))) (+ 1.0 (* -0.16666666666666666 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00031) {
		tmp = sin(y) / y;
	} else if (x <= 4e+153) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (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) :: tmp
    if (x <= 0.00031d0) then
        tmp = sin(y) / y
    else if (x <= 4d+153) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00031) {
		tmp = Math.sin(y) / y;
	} else if (x <= 4e+153) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00031:
		tmp = math.sin(y) / y
	elif x <= 4e+153:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(sin(y) / y);
	elseif (x <= 4e+153)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = sin(y) / y;
	elseif (x <= 4e+153)
		tmp = cosh(x);
	else
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00031], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4e+153], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1e-4

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 67.6%

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

   if 3.1e-4 < x < 4e153

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 90.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   if 4e153 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 84.2%

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

      1. unpow2 18.3%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in x around 0 84.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 68.4%

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

   7. Simplified 84.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.2e+153)
   (cosh x)
   (* (+ 1.0 (* 0.5 (* x x))) (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+153) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (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) :: tmp
    if (x <= 6.2d+153) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.2e+153) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.2e+153:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.2e+153)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.2e+153)
		tmp = cosh(x);
	else
		tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.2e+153], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2e153

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 62.9%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   if 6.2e153 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 84.2%

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

      1. unpow2 18.3%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in x around 0 84.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 68.4%

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

   7. Simplified 84.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 48.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (+ 1.0 (* 0.5 (* x x))) (+ 1.0 (* -0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (-0.16666666666666666 * (y * y)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in y around 0 65.4%

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

   1. unpow2 35.6%

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

4. Simplified 65.4%

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

5. Taylor expanded in x around 0 53.6%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]
6. Step-by-step derivation

   1. unpow2 46.0%

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

7. Simplified 53.6%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

8. Final simplification 53.6%

   \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 8: 48.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (+ 1.0 (* 0.5 (* x x))) (+ 1.0 (* y (* y -0.16666666666666666)))))

C

double code(double x, double y) {
	return (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * -0.16666666666666666)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (0.5d0 * (x * x))) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * -0.16666666666666666)));
}

Python

def code(x, y):
	return (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * -0.16666666666666666)))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (0.5 * (x * x))) * (1.0 + (y * (y * -0.16666666666666666)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 74.1%

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

   2. pow2 74.1%

      \[\leadsto \cosh x \cdot \color{blue}{{\left(\sqrt{\frac{\sin y}{y}}\right)}^{2}} \]

3. Applied egg-rr 74.1%

   \[\leadsto \cosh x \cdot \color{blue}{{\left(\sqrt{\frac{\sin y}{y}}\right)}^{2}} \]

4. Taylor expanded in x around 0 55.9%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot {\left(\sqrt{\frac{\sin y}{y}}\right)}^{2} \]
5. Step-by-step derivation

   1. unpow2 46.0%

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

6. Simplified 55.9%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot x\right)\right)} \cdot {\left(\sqrt{\frac{\sin y}{y}}\right)}^{2} \]

7. Taylor expanded in y around 0 53.6%

   \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
8. Step-by-step derivation

   1. *-commutative 53.6%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666}\right) \]

   2. unpow2 53.6%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + \color{blue}{\left(y \cdot y\right)} \cdot -0.16666666666666666\right) \]

   3. associate-*r* 53.6%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + \color{blue}{y \cdot \left(y \cdot -0.16666666666666666\right)}\right) \]

9. Simplified 53.6%

   \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)} \]

10. Final simplification 53.6%

    \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

Alternative 9: 39.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e+155)
   (+ 1.0 (* y (* y -0.16666666666666666)))
   (+ 1.0 (* 0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e+155) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 1.0 + (0.5 * (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 (x <= 1.8d+155) then
        tmp = 1.0d0 + (y * (y * (-0.16666666666666666d0)))
    else
        tmp = 1.0d0 + (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.8e+155) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 1.0 + (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.8e+155:
		tmp = 1.0 + (y * (y * -0.16666666666666666))
	else:
		tmp = 1.0 + (0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e+155)
		tmp = Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.8e+155)
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	else
		tmp = 1.0 + (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e+155], N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004e155

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 58.7%

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

   3. Taylor expanded in y around 0 38.9%

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

      1. unpow2 38.9%

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

   5. Simplified 38.9%

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

   6. Taylor expanded in y around 0 38.9%

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

      1. *-commutative 38.9%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 38.9%

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

      3. associate-*r* 38.9%

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

   8. Simplified 38.9%

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

   if 1.80000000000000004e155 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 70.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]

   3. Taylor expanded in x around 0 70.3%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)} \cdot 1 \]
   4. Step-by-step derivation

      1. unpow2 70.3%

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

   5. Simplified 70.3%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 10: 31.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 50.6%

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

3. Taylor expanded in y around 0 35.6%

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

   1. unpow2 35.6%

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

5. Simplified 35.6%

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

6. Final simplification 35.6%

   \[\leadsto 1 + -0.16666666666666666 \cdot \left(y \cdot y\right) \]

Alternative 11: 31.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 50.6%

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

3. Taylor expanded in y around 0 35.6%

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

   1. unpow2 35.6%

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

5. Simplified 35.6%

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

6. Taylor expanded in y around 0 35.6%

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

   1. *-commutative 35.6%

      \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

   2. unpow2 35.6%

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

   3. associate-*r* 35.6%

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

8. Simplified 35.6%

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

9. Final simplification 35.6%

   \[\leadsto 1 + y \cdot \left(y \cdot -0.16666666666666666\right) \]

Alternative 12: 26.6% 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 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 50.6%

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

3. Taylor expanded in y around 0 29.3%

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

4. Final simplification 29.3%

   \[\leadsto 1 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))