Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 12

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 86.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.0) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x) <= 1.0d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 1 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.6%

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

      1. unpow2 17.8%

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

   4. Simplified 78.6%

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

4. Final simplification 87.5%

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

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.0) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.0:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.0)
		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) <= 1.0)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   if 1 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 73.2%

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

4. Final simplification 84.3%

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

Alternative 4: 72.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y))
        (t_1 (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))))
   (if (<= x 2.05e-7)
     t_0
     (if (<= x 2e+70)
       t_1
       (if (<= x 2.5e+135)
         (cosh x)
         (if (<= x 1.35e+154) t_1 (* t_0 (* 0.5 (* x x)))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	double tmp;
	if (x <= 2.05e-7) {
		tmp = t_0;
	} else if (x <= 2e+70) {
		tmp = t_1;
	} else if (x <= 2.5e+135) {
		tmp = cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) / y
    t_1 = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    if (x <= 2.05d-7) then
        tmp = t_0
    else if (x <= 2d+70) then
        tmp = t_1
    else if (x <= 2.5d+135) then
        tmp = cosh(x)
    else if (x <= 1.35d+154) then
        tmp = t_1
    else
        tmp = t_0 * (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	double tmp;
	if (x <= 2.05e-7) {
		tmp = t_0;
	} else if (x <= 2e+70) {
		tmp = t_1;
	} else if (x <= 2.5e+135) {
		tmp = Math.cosh(x);
	} else if (x <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0 * (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	tmp = 0
	if x <= 2.05e-7:
		tmp = t_0
	elif x <= 2e+70:
		tmp = t_1
	elif x <= 2.5e+135:
		tmp = math.cosh(x)
	elif x <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_0 * (0.5 * (x * x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))))
	tmp = 0.0
	if (x <= 2.05e-7)
		tmp = t_0;
	elseif (x <= 2e+70)
		tmp = t_1;
	elseif (x <= 2.5e+135)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	tmp = 0.0;
	if (x <= 2.05e-7)
		tmp = t_0;
	elseif (x <= 2e+70)
		tmp = t_1;
	elseif (x <= 2.5e+135)
		tmp = cosh(x);
	elseif (x <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0 * (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.05e-7], t$95$0, If[LessEqual[x, 2e+70], t$95$1, If[LessEqual[x, 2.5e+135], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 1.35e+154], t$95$1, N[(t$95$0 * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.05e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 60.9%

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

   if 2.05e-7 < x < 2.00000000000000015e70 or 2.50000000000000015e135 < x < 1.35000000000000003e154

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 82.5%

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

      1. unpow2 31.2%

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

   4. Simplified 82.5%

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

   if 2.00000000000000015e70 < x < 2.50000000000000015e135

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.6%

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

   if 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 100.0%

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

   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. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow2 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+242}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e+242)
   (cosh x)
   (* (+ 1.0 (* -0.16666666666666666 (* y y))) (+ 1.0 (* 0.5 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e+242) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (0.5 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d+242) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * (1.0d0 + (0.5d0 * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e+242) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (0.5 * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e+242:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (0.5 * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e+242)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e+242)
		tmp = cosh(x);
	else
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + (0.5 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e+242], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000003e242

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 65.1%

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

   if 8.5000000000000003e242 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 56.8%

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

      1. unpow2 56.8%

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

   4. Simplified 56.8%

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

   5. Taylor expanded in x around 0 56.8%

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

      1. unpow2 63.4%

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

   7. Simplified 56.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+242}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 6: 47.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;1 + t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x x))))
   (if (<= y 14500000000.0)
     (+ 1.0 t_0)
     (* (+ 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 (y <= 14500000000.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * 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 (y <= 14500000000.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * 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 (y <= 14500000000.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x * x)
	tmp = 0
	if y <= 14500000000.0:
		tmp = 1.0 + t_0
	else:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x * x))
	tmp = 0.0
	if (y <= 14500000000.0)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x * x);
	tmp = 0.0;
	if (y <= 14500000000.0)
		tmp = 1.0 + t_0;
	else
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * 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[y, 14500000000.0], N[(1.0 + t$95$0), $MachinePrecision], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e10

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 73.1%

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

   3. Taylor expanded in x around 0 52.8%

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

      1. unpow2 73.8%

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

   5. Simplified 52.8%

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

   if 1.45e10 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. unpow2 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/l* 28.2%

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

      3. associate-/r/ 28.2%

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

      4. associate-*r* 28.2%

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

      5. *-commutative 28.2%

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

      6. unpow2 28.2%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in y around 0 36.4%

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

      1. unpow2 27.7%

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

   10. Simplified 36.4%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 7: 49.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 64.7%

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

   1. unpow2 29.2%

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

4. Simplified 64.7%

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

5. Taylor expanded in x around 0 49.2%

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

   1. unpow2 72.5%

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

7. Simplified 49.2%

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

8. Final simplification 49.2%

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

Alternative 8: 47.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 14500000000.0)
   (+ 1.0 (* 0.5 (* x x)))
   (* (* x x) (+ 0.5 (* (* y y) -0.08333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 14500000000.0) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = (x * x) * (0.5 + ((y * y) * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 14500000000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * x))
    else
        tmp = (x * x) * (0.5d0 + ((y * y) * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 14500000000.0) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = (x * x) * (0.5 + ((y * y) * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 14500000000.0:
		tmp = 1.0 + (0.5 * (x * x))
	else:
		tmp = (x * x) * (0.5 + ((y * y) * -0.08333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 14500000000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(y * y) * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 14500000000.0)
		tmp = 1.0 + (0.5 * (x * x));
	else
		tmp = (x * x) * (0.5 + ((y * y) * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 14500000000.0], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(y * y), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e10

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 73.1%

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

   3. Taylor expanded in x around 0 52.8%

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

      1. unpow2 73.8%

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

   5. Simplified 52.8%

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

   if 1.45e10 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 67.7%

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

      1. unpow2 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in x around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/l* 28.2%

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

      3. associate-/r/ 28.2%

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

      4. associate-*r* 28.2%

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

      5. *-commutative 28.2%

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

      6. unpow2 28.2%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in y around 0 20.6%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{2} + -0.08333333333333333 \cdot \left({y}^{2} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 20.6%

         \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left({y}^{2} \cdot {x}^{2}\right) + 0.5 \cdot {x}^{2}} \]

      2. associate-*r* 20.6%

         \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot {y}^{2}\right) \cdot {x}^{2}} + 0.5 \cdot {x}^{2} \]

      3. distribute-rgt-out 36.4%

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(-0.08333333333333333 \cdot {y}^{2} + 0.5\right)} \]

      4. unpow2 36.4%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(-0.08333333333333333 \cdot {y}^{2} + 0.5\right) \]

      5. *-commutative 36.4%

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{{y}^{2} \cdot -0.08333333333333333} + 0.5\right) \]

      6. unpow2 36.4%

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot -0.08333333333333333 + 0.5\right) \]

   10. Simplified 36.4%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(y \cdot y\right) \cdot -0.08333333333333333 + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 14500000000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + \left(y \cdot y\right) \cdot -0.08333333333333333\right)\\ \end{array} \]

Alternative 9: 40.9% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+153}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e+153) (+ 1.0 (* -0.16666666666666666 (* y y))) (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e+153) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} 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 <= 6d+153) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    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 <= 6e+153) {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e+153:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e+153)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(0.5 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e+153)
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e+153], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000037e153

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 50.8%

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

   3. Taylor expanded in y around 0 30.7%

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

      1. unpow2 30.7%

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

   5. Simplified 30.7%

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

   if 6.00000000000000037e153 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.8%

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

      1. unpow2 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-/l* 100.0%

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

      3. associate-/r/ 97.8%

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

      4. associate-*r* 97.8%

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

      5. *-commutative 97.8%

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

      6. unpow2 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in y around 0 78.8%

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

      1. unpow2 78.8%

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

   10. Simplified 78.8%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+153}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 10: 46.9% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.25e+157)
   (+ 1.0 (* 0.5 (* x x)))
   (+ 1.0 (* -0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+157) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.25d+157) then
        tmp = 1.0d0 + (0.5d0 * (x * x))
    else
        tmp = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+157) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.25e+157:
		tmp = 1.0 + (0.5 * (x * x))
	else:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.25e+157)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.25e+157)
		tmp = 1.0 + (0.5 * (x * x));
	else
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.25e+157], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.24999999999999994e157

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 67.9%

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

   3. Taylor expanded in x around 0 48.1%

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

      1. unpow2 73.3%

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

   5. Simplified 48.1%

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

   if 1.24999999999999994e157 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 35.4%

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

   3. Taylor expanded in y around 0 47.5%

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

      1. unpow2 47.5%

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

   5. Simplified 47.5%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 11: 36.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.4) 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4) {
		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 <= 1.4d0) 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 <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4)
		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 <= 1.4)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4], 1.0, N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 61.0%

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

   3. Taylor expanded in y around 0 28.9%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 50.2%

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

      1. unpow2 50.2%

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

   4. Simplified 50.2%

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

   5. Taylor expanded in x around inf 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-/l* 57.4%

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

      3. associate-/r/ 50.2%

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

      4. associate-*r* 50.2%

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

      5. *-commutative 50.2%

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

      6. unpow2 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in y around 0 40.4%

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

      1. unpow2 40.4%

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

   10. Simplified 40.4%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 12: 27.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 43.8%

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

3. Taylor expanded in y around 0 21.1%

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

4. Final simplification 21.1%

   \[\leadsto 1 \]

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 2023221 
(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)))