Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 13

Speedup: 1.0×

• 48.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-*l/ 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 10:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 10.0) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 10.0) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x) <= 10.0d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 10.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 10.0:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 10.0)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 10.0)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 10.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 10

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.4%

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

   if 10 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.4%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 10:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 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 4: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (sin y) (/ y (* x x))))))
   (if (<= x -1.35e+154)
     t_0
     (if (<= x -92000000.0)
       (cosh x)
       (if (<= x 0.37) (/ (sin y) y) (if (<= x 3.2e+107) (cosh x) t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = cosh(x);
	} else if (x <= 0.37) {
		tmp = sin(y) / y;
	} else if (x <= 3.2e+107) {
		tmp = cosh(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 = 0.5d0 * (sin(y) / (y / (x * x)))
    if (x <= (-1.35d+154)) then
        tmp = t_0
    else if (x <= (-92000000.0d0)) then
        tmp = cosh(x)
    else if (x <= 0.37d0) then
        tmp = sin(y) / y
    else if (x <= 3.2d+107) then
        tmp = cosh(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (Math.sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = Math.cosh(x);
	} else if (x <= 0.37) {
		tmp = Math.sin(y) / y;
	} else if (x <= 3.2e+107) {
		tmp = Math.cosh(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sin(y) / (y / (x * x)))
	tmp = 0
	if x <= -1.35e+154:
		tmp = t_0
	elif x <= -92000000.0:
		tmp = math.cosh(x)
	elif x <= 0.37:
		tmp = math.sin(y) / y
	elif x <= 3.2e+107:
		tmp = math.cosh(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = Float64(sin(y) / y);
	elseif (x <= 3.2e+107)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (sin(y) / (y / (x * x)));
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = sin(y) / y;
	elseif (x <= 3.2e+107)
		tmp = cosh(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], t$95$0, If[LessEqual[x, -92000000.0], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 0.37], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 3.2e+107], N[Cosh[x], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154 or 3.20000000000000029e107 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.8%

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

      1. unpow2 69.2%

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

   4. Simplified 85.8%

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

   5. Taylor expanded in x around inf 85.8%

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

      1. associate-/l* 92.2%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\sin y}{\frac{y}{{x}^{2}}}} \]

      2. unpow2 92.2%

         \[\leadsto 0.5 \cdot \frac{\sin y}{\frac{y}{\color{blue}{x \cdot x}}} \]

   7. Simplified 92.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}} \]

   if -1.35000000000000003e154 < x < -9.2e7 or 0.37 < x < 3.20000000000000029e107

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.0%

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

   if -9.2e7 < x < 0.37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 5: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (sin y) (/ y (* x x))))))
   (if (<= x -1.35e+154)
     t_0
     (if (<= x -92000000.0)
       (cosh x)
       (if (<= x 0.37)
         (/ (sin y) y)
         (if (<= x 1.35e+154)
           (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
           t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = cosh(x);
	} else if (x <= 0.37) {
		tmp = sin(y) / y;
	} else if (x <= 1.35e+154) {
		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 = 0.5d0 * (sin(y) / (y / (x * x)))
    if (x <= (-1.35d+154)) then
        tmp = t_0
    else if (x <= (-92000000.0d0)) then
        tmp = cosh(x)
    else if (x <= 0.37d0) then
        tmp = sin(y) / y
    else if (x <= 1.35d+154) 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 = 0.5 * (Math.sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = Math.cosh(x);
	} else if (x <= 0.37) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1.35e+154) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sin(y) / (y / (x * x)))
	tmp = 0
	if x <= -1.35e+154:
		tmp = t_0
	elif x <= -92000000.0:
		tmp = math.cosh(x)
	elif x <= 0.37:
		tmp = math.sin(y) / y
	elif x <= 1.35e+154:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1.35e+154)
		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 = 0.5 * (sin(y) / (y / (x * x)));
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = sin(y) / y;
	elseif (x <= 1.35e+154)
		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[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], t$95$0, If[LessEqual[x, -92000000.0], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 0.37], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.35e+154], 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 4 regimes

2. if x < -1.35000000000000003e154 or 1.35000000000000003e154 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 80.8%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\sin y}{\frac{y}{{x}^{2}}}} \]

      2. unpow2 100.0%

         \[\leadsto 0.5 \cdot \frac{\sin y}{\frac{y}{\color{blue}{x \cdot x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}} \]

   if -1.35000000000000003e154 < x < -9.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 92.3%

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

   if -9.2e7 < x < 0.37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.0%

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

   if 0.37 < x < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.2%

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

      1. unpow2 40.4%

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

   4. Simplified 85.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 6: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ (sin y) (/ y (* x x))))))
   (if (<= x -1.35e+154)
     t_0
     (if (<= x -92000000.0)
       (cosh x)
       (if (<= x 0.37)
         (* (/ (sin y) y) (+ 1.0 (* 0.5 (* x x))))
         (if (<= x 1.35e+154)
           (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
           t_0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = cosh(x);
	} else if (x <= 0.37) {
		tmp = (sin(y) / y) * (1.0 + (0.5 * (x * x)));
	} else if (x <= 1.35e+154) {
		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 = 0.5d0 * (sin(y) / (y / (x * x)))
    if (x <= (-1.35d+154)) then
        tmp = t_0
    else if (x <= (-92000000.0d0)) then
        tmp = cosh(x)
    else if (x <= 0.37d0) then
        tmp = (sin(y) / y) * (1.0d0 + (0.5d0 * (x * x)))
    else if (x <= 1.35d+154) 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 = 0.5 * (Math.sin(y) / (y / (x * x)));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = t_0;
	} else if (x <= -92000000.0) {
		tmp = Math.cosh(x);
	} else if (x <= 0.37) {
		tmp = (Math.sin(y) / y) * (1.0 + (0.5 * (x * x)));
	} else if (x <= 1.35e+154) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sin(y) / (y / (x * x)))
	tmp = 0
	if x <= -1.35e+154:
		tmp = t_0
	elif x <= -92000000.0:
		tmp = math.cosh(x)
	elif x <= 0.37:
		tmp = (math.sin(y) / y) * (1.0 + (0.5 * (x * x)))
	elif x <= 1.35e+154:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = Float64(Float64(sin(y) / y) * Float64(1.0 + Float64(0.5 * Float64(x * x))));
	elseif (x <= 1.35e+154)
		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 = 0.5 * (sin(y) / (y / (x * x)));
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = t_0;
	elseif (x <= -92000000.0)
		tmp = cosh(x);
	elseif (x <= 0.37)
		tmp = (sin(y) / y) * (1.0 + (0.5 * (x * x)));
	elseif (x <= 1.35e+154)
		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[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], t$95$0, If[LessEqual[x, -92000000.0], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 0.37], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], 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 4 regimes

2. if x < -1.35000000000000003e154 or 1.35000000000000003e154 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 80.8%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. associate-/l* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\sin y}{\frac{y}{{x}^{2}}}} \]

      2. unpow2 100.0%

         \[\leadsto 0.5 \cdot \frac{\sin y}{\frac{y}{\color{blue}{x \cdot x}}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}} \]

   if -1.35000000000000003e154 < x < -9.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 92.3%

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

   if -9.2e7 < x < 0.37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.2%

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

      1. unpow2 44.6%

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

   4. Simplified 97.2%

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

   if 0.37 < x < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 85.2%

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

      1. unpow2 40.4%

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

   4. Simplified 85.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;x \leq -92000000:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 7: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+233}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4e+233) (+ 1.0 (* y (* y -0.16666666666666666))) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+233) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = cosh(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 <= (-3.4d+233)) then
        tmp = 1.0d0 + (y * (y * (-0.16666666666666666d0)))
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4e+233) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4e+233:
		tmp = 1.0 + (y * (y * -0.16666666666666666))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4e+233)
		tmp = Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.4e+233)
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4e+233], N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.40000000000000022e233

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 67.3%

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

      1. unpow2 2.4%

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

   4. Simplified 67.3%

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

      1. add-cube-cbrt 66.7%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 0.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in x around 0 0.0%

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

   10. Taylor expanded in y around 0 39.7%

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

       1. *-commutative 39.7%

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

       2. unpow2 39.7%

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

       3. associate-*l* 39.7%

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

   12. Simplified 39.7%

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

   if -3.40000000000000022e233 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 65.0%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+233}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 8: 47.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;1 + t_0\\ \mathbf{elif}\;x \leq 10^{+154} \lor \neg \left(x \leq 5 \cdot 10^{+201}\right):\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x x))))
   (if (<= x -4.4e-20)
     (* 0.5 (/ (* y (* x x)) y))
     (if (<= x 1.4e+54)
       (+ 1.0 t_0)
       (if (or (<= x 1e+154) (not (<= x 5e+201)))
         (* 0.5 (* (* x x) (+ 1.0 (* -0.16666666666666666 (* y y)))))
         t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x * x);
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 1.4e+54) {
		tmp = 1.0 + t_0;
	} else if ((x <= 1e+154) || !(x <= 5e+201)) {
		tmp = 0.5 * ((x * 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 = 0.5d0 * (x * x)
    if (x <= (-4.4d-20)) then
        tmp = 0.5d0 * ((y * (x * x)) / y)
    else if (x <= 1.4d+54) then
        tmp = 1.0d0 + t_0
    else if ((x <= 1d+154) .or. (.not. (x <= 5d+201))) then
        tmp = 0.5d0 * ((x * 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 = 0.5 * (x * x);
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 1.4e+54) {
		tmp = 1.0 + t_0;
	} else if ((x <= 1e+154) || !(x <= 5e+201)) {
		tmp = 0.5 * ((x * x) * (1.0 + (-0.16666666666666666 * (y * y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x * x)
	tmp = 0
	if x <= -4.4e-20:
		tmp = 0.5 * ((y * (x * x)) / y)
	elif x <= 1.4e+54:
		tmp = 1.0 + t_0
	elif (x <= 1e+154) or not (x <= 5e+201):
		tmp = 0.5 * ((x * x) * (1.0 + (-0.16666666666666666 * (y * y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (x <= -4.4e-20)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(x * x)) / y));
	elseif (x <= 1.4e+54)
		tmp = Float64(1.0 + t_0);
	elseif ((x <= 1e+154) || !(x <= 5e+201))
		tmp = Float64(0.5 * Float64(Float64(x * 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 = 0.5 * (x * x);
	tmp = 0.0;
	if (x <= -4.4e-20)
		tmp = 0.5 * ((y * (x * x)) / y);
	elseif (x <= 1.4e+54)
		tmp = 1.0 + t_0;
	elseif ((x <= 1e+154) || ~((x <= 5e+201)))
		tmp = 0.5 * ((x * 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[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-20], N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e+54], N[(1.0 + t$95$0), $MachinePrecision], If[Or[LessEqual[x, 1e+154], N[Not[LessEqual[x, 5e+201]], $MachinePrecision]], N[(0.5 * N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.39999999999999982e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 48.6%

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

   4. Simplified 61.1%

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

      1. add-cube-cbrt 61.1%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 33.2%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf 58.6%

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

       1. unpow2 58.6%

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

   11. Simplified 58.6%

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

   12. Taylor expanded in y around 0 56.5%

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

       1. unpow2 56.5%

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

   14. Simplified 56.5%

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

   if -4.39999999999999982e-20 < x < 1.40000000000000008e54

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 48.3%

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

   3. Taylor expanded in x around 0 44.0%

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

      1. unpow2 44.0%

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

   5. Simplified 44.0%

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

   if 1.40000000000000008e54 < x < 1.00000000000000004e154 or 4.9999999999999995e201 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.0%

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

      1. unpow2 38.8%

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

   4. Simplified 56.0%

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

      1. add-cube-cbrt 56.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 19.9%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 19.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in x around inf 56.0%

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

       1. unpow2 56.0%

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

   11. Simplified 56.0%

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

   12. Taylor expanded in y around 0 24.8%

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

       1. unpow2 24.8%

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

       2. associate-*r* 24.8%

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

       3. unpow2 24.8%

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

       4. distribute-lft1-in 68.9%

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

       5. *-commutative 68.9%

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

       6. unpow2 68.9%

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

   14. Simplified 68.9%

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

   if 1.00000000000000004e154 < x < 4.9999999999999995e201

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 40.0%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 40.0%

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

   8. Simplified 40.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 100.0%

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

       1. unpow2 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+54}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 10^{+154} \lor \neg \left(x \leq 5 \cdot 10^{+201}\right):\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 9: 47.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot x\right)\\ t_1 := 1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+155}:\\ \;\;\;\;\left(1 + t_0\right) \cdot t_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+201}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x x))) (t_1 (+ 1.0 (* -0.16666666666666666 (* y y)))))
   (if (<= x -4.4e-20)
     (* 0.5 (/ (* y (* x x)) y))
     (if (<= x 6.6e+155)
       (* (+ 1.0 t_0) t_1)
       (if (<= x 5e+201) t_0 (* 0.5 (* (* x x) t_1)))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x * x);
	double t_1 = 1.0 + (-0.16666666666666666 * (y * y));
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 6.6e+155) {
		tmp = (1.0 + t_0) * t_1;
	} else if (x <= 5e+201) {
		tmp = t_0;
	} else {
		tmp = 0.5 * ((x * x) * t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (x * x)
    t_1 = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    if (x <= (-4.4d-20)) then
        tmp = 0.5d0 * ((y * (x * x)) / y)
    else if (x <= 6.6d+155) then
        tmp = (1.0d0 + t_0) * t_1
    else if (x <= 5d+201) then
        tmp = t_0
    else
        tmp = 0.5d0 * ((x * x) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x * x);
	double t_1 = 1.0 + (-0.16666666666666666 * (y * y));
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 6.6e+155) {
		tmp = (1.0 + t_0) * t_1;
	} else if (x <= 5e+201) {
		tmp = t_0;
	} else {
		tmp = 0.5 * ((x * x) * t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x * x)
	t_1 = 1.0 + (-0.16666666666666666 * (y * y))
	tmp = 0
	if x <= -4.4e-20:
		tmp = 0.5 * ((y * (x * x)) / y)
	elif x <= 6.6e+155:
		tmp = (1.0 + t_0) * t_1
	elif x <= 5e+201:
		tmp = t_0
	else:
		tmp = 0.5 * ((x * x) * t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x * x))
	t_1 = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))
	tmp = 0.0
	if (x <= -4.4e-20)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(x * x)) / y));
	elseif (x <= 6.6e+155)
		tmp = Float64(Float64(1.0 + t_0) * t_1);
	elseif (x <= 5e+201)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(Float64(x * x) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x * x);
	t_1 = 1.0 + (-0.16666666666666666 * (y * y));
	tmp = 0.0;
	if (x <= -4.4e-20)
		tmp = 0.5 * ((y * (x * x)) / y);
	elseif (x <= 6.6e+155)
		tmp = (1.0 + t_0) * t_1;
	elseif (x <= 5e+201)
		tmp = t_0;
	else
		tmp = 0.5 * ((x * x) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-20], N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+155], N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x, 5e+201], t$95$0, N[(0.5 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.39999999999999982e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 48.6%

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

   4. Simplified 61.1%

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

      1. add-cube-cbrt 61.1%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 33.2%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf 58.6%

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

       1. unpow2 58.6%

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

   11. Simplified 58.6%

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

   12. Taylor expanded in y around 0 56.5%

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

       1. unpow2 56.5%

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

   14. Simplified 56.5%

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

   if -4.39999999999999982e-20 < x < 6.5999999999999997e155

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 82.5%

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

      1. unpow2 38.7%

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

   4. Simplified 82.5%

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

   5. Taylor expanded in y around 0 44.4%

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

      1. unpow2 44.4%

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

   7. Simplified 44.4%

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

   if 6.5999999999999997e155 < x < 4.9999999999999995e201

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 40.0%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 40.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 40.0%

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

   8. Simplified 40.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 100.0%

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

       1. unpow2 100.0%

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

   14. Simplified 100.0%

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

   if 4.9999999999999995e201 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 69.6%

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

   4. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 34.8%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 34.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 34.8%

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

   8. Simplified 34.8%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 0.0%

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

       1. unpow2 0.0%

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

       2. associate-*r* 0.0%

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

       3. unpow2 0.0%

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

       4. distribute-lft1-in 82.6%

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

       5. *-commutative 82.6%

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

       6. unpow2 82.6%

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

   14. Simplified 82.6%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\left(y \cdot y\right) \cdot -0.16666666666666666 + 1\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+155}:\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+201}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot x\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]

Alternative 10: 46.9% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-20)
   (* 0.5 (/ (* y (* x x)) y))
   (if (<= x 1.26e+154)
     (+ 1.0 (* y (* y -0.16666666666666666)))
     (* 0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 1.26e+154) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 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 <= (-4.4d-20)) then
        tmp = 0.5d0 * ((y * (x * x)) / y)
    else if (x <= 1.26d+154) then
        tmp = 1.0d0 + (y * (y * (-0.16666666666666666d0)))
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 1.26e+154) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-20:
		tmp = 0.5 * ((y * (x * x)) / y)
	elif x <= 1.26e+154:
		tmp = 1.0 + (y * (y * -0.16666666666666666))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-20)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(x * x)) / y));
	elseif (x <= 1.26e+154)
		tmp = Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-20)
		tmp = 0.5 * ((y * (x * x)) / y);
	elseif (x <= 1.26e+154)
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-20], N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.26e+154], N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.39999999999999982e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 48.6%

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

   4. Simplified 61.1%

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

      1. add-cube-cbrt 61.1%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 33.2%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf 58.6%

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

       1. unpow2 58.6%

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

   11. Simplified 58.6%

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

   12. Taylor expanded in y around 0 56.5%

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

       1. unpow2 56.5%

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

   14. Simplified 56.5%

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

   if -4.39999999999999982e-20 < x < 1.26e154

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 82.5%

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

      1. unpow2 38.7%

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

   4. Simplified 82.5%

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

      1. add-cube-cbrt 80.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 37.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 37.3%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 37.3%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 37.3%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 37.3%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 37.3%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 37.3%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 37.8%

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

   8. Simplified 37.8%

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

   9. Taylor expanded in x around 0 37.5%

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

   10. Taylor expanded in y around 0 41.6%

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

       1. *-commutative 41.6%

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

       2. unpow2 41.6%

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

       3. associate-*l* 41.6%

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

   12. Simplified 41.6%

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

   if 1.26e154 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 78.8%

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

   4. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 36.4%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 36.4%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 36.4%

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

   8. Simplified 36.4%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around 0 78.8%

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

       1. unpow2 78.8%

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

   14. Simplified 78.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 11: 45.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-20)
   (* 0.5 (/ (* y (* x x)) y))
   (if (<= x 6.2e-10) 1.0 (* 0.5 (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 6.2e-10) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-4.4d-20)) then
        tmp = 0.5d0 * ((y * (x * x)) / y)
    else if (x <= 6.2d-10) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-20) {
		tmp = 0.5 * ((y * (x * x)) / y);
	} else if (x <= 6.2e-10) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-20:
		tmp = 0.5 * ((y * (x * x)) / y)
	elif x <= 6.2e-10:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-20)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(x * x)) / y));
	elseif (x <= 6.2e-10)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-20)
		tmp = 0.5 * ((y * (x * x)) / y);
	elseif (x <= 6.2e-10)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-20], N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-10], 1.0, N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.39999999999999982e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 48.6%

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

   4. Simplified 61.1%

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

      1. add-cube-cbrt 61.1%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 33.2%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 33.2%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 33.3%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around inf 58.6%

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

       1. unpow2 58.6%

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

   11. Simplified 58.6%

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

   12. Taylor expanded in y around 0 56.5%

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

       1. unpow2 56.5%

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

   14. Simplified 56.5%

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

   if -4.39999999999999982e-20 < x < 6.2000000000000003e-10

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 46.9%

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

   3. Taylor expanded in x around 0 46.9%

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

      1. unpow2 46.9%

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around 0 46.9%

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

   if 6.2000000000000003e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 57.6%

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

      1. unpow2 43.7%

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

   4. Simplified 57.6%

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

      1. add-cube-cbrt 57.6%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 20.9%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 20.9%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 20.9%

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

   8. Simplified 20.9%

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

   9. Taylor expanded in x around inf 55.9%

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

       1. unpow2 55.9%

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

   11. Simplified 55.9%

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

   12. Taylor expanded in y around 0 43.7%

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

       1. unpow2 43.7%

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

   14. Simplified 43.7%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 12: 44.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20} \lor \neg \left(x \leq 6.2 \cdot 10^{-10}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.4e-20) (not (<= x 6.2e-10))) (* 0.5 (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.4e-20) || !(x <= 6.2e-10)) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4.4d-20)) .or. (.not. (x <= 6.2d-10))) then
        tmp = 0.5d0 * (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.4e-20) || !(x <= 6.2e-10)) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.4e-20) or not (x <= 6.2e-10):
		tmp = 0.5 * (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.4e-20) || !(x <= 6.2e-10))
		tmp = Float64(0.5 * Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.4e-20) || ~((x <= 6.2e-10)))
		tmp = 0.5 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.4e-20], N[Not[LessEqual[x, 6.2e-10]], $MachinePrecision]], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999982e-20 or 6.2000000000000003e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 59.5%

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

      1. unpow2 46.3%

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

   4. Simplified 59.5%

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

      1. add-cube-cbrt 59.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}}{y} \]

      2. add-sqr-sqrt 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}} \]

      3. times-frac 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]

      4. pow2 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{\sin y}\right)}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right) \]

   6. Applied egg-rr 27.5%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sin y}\right)}^{2}}{\sqrt{y}} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \frac{\sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}}} \]

      2. associate-*r/ 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin y}\right)}^{2} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}}{\sqrt{y}} \]

      3. unpow2 27.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right)} \cdot \sqrt[3]{\sin y}}{\sqrt{y}}}{\sqrt{y}} \]

      4. rem-3cbrt-lft 27.5%

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

   8. Simplified 27.5%

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

   9. Taylor expanded in x around inf 57.3%

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

       1. unpow2 57.3%

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

   11. Simplified 57.3%

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

   12. Taylor expanded in y around 0 46.3%

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

       1. unpow2 46.3%

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

   14. Simplified 46.3%

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

   if -4.39999999999999982e-20 < x < 6.2000000000000003e-10

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 46.9%

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

   3. Taylor expanded in x around 0 46.9%

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

      1. unpow2 46.9%

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around 0 46.9%

      \[\leadsto \color{blue}{1} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-20} \lor \neg \left(x \leq 6.2 \cdot 10^{-10}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 27.1% 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 y around 0 61.3%

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

3. Taylor expanded in x around 0 46.6%

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

   1. unpow2 46.6%

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

5. Simplified 46.6%

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

6. Taylor expanded in x around 0 24.0%

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

7. Final simplification 24.0%

   \[\leadsto 1 \]

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 2023181 
(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)))