Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 8

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: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000000000001:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.000000000001) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.000000000001) {
		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) <= 1.000000000001d0) 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) <= 1.000000000001) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.000000000001:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.000000000001)
		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) <= 1.000000000001)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.000000000001], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.0000000000010001

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.6%

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

   if 1.0000000000010001 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 76.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000000000001:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 10^{+58}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+147}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\left(x \cdot x\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.6e-6)
   (/ (sin y) y)
   (if (<= x 1e+58)
     (cosh x)
     (if (<= x 7.5e+147)
       (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
       (* (sin y) (* (* x x) (/ 0.5 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.6e-6) {
		tmp = sin(y) / y;
	} else if (x <= 1e+58) {
		tmp = cosh(x);
	} else if (x <= 7.5e+147) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = sin(y) * ((x * x) * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.6d-6) then
        tmp = sin(y) / y
    else if (x <= 1d+58) then
        tmp = cosh(x)
    else if (x <= 7.5d+147) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = sin(y) * ((x * x) * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.6e-6) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1e+58) {
		tmp = Math.cosh(x);
	} else if (x <= 7.5e+147) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * ((x * x) * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.6e-6:
		tmp = math.sin(y) / y
	elif x <= 1e+58:
		tmp = math.cosh(x)
	elif x <= 7.5e+147:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * ((x * x) * (0.5 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.6e-6)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1e+58)
		tmp = cosh(x);
	elseif (x <= 7.5e+147)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sin(y) * Float64(Float64(x * x) * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.6e-6)
		tmp = sin(y) / y;
	elseif (x <= 1e+58)
		tmp = cosh(x);
	elseif (x <= 7.5e+147)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * ((x * x) * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.6e-6], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1e+58], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 7.5e+147], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.5999999999999999e-6

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 62.1%

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

   if 1.5999999999999999e-6 < x < 9.99999999999999944e57

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 87.3%

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

   if 9.99999999999999944e57 < x < 7.50000000000000037e147

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 87.5%

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

      1. unpow2 33.0%

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

   4. Simplified 87.5%

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

   if 7.50000000000000037e147 < x

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

       2. associate-*l/ 71.3%

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

       3. *-commutative 71.3%

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

       4. associate-*l/ 100.0%

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

       5. unpow2 100.0%

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

       6. *-lft-identity 100.0%

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

       7. associate-*l/ 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-*r* 100.0%

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

       10. unpow2 100.0%

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

       11. associate-*l/ 100.0%

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

       12. metadata-eval 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 10^{+58}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+147}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\left(x \cdot x\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.6:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+147}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\left(x \cdot x\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.6)
   (/ (+ 1.0 (* 0.5 (* x x))) (/ y (sin y)))
   (if (<= x 1.1e+64)
     (cosh x)
     (if (<= x 7.5e+147)
       (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
       (* (sin y) (* (* x x) (/ 0.5 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.6) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	} else if (x <= 1.1e+64) {
		tmp = cosh(x);
	} else if (x <= 7.5e+147) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = sin(y) * ((x * x) * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9.6d0) then
        tmp = (1.0d0 + (0.5d0 * (x * x))) / (y / sin(y))
    else if (x <= 1.1d+64) then
        tmp = cosh(x)
    else if (x <= 7.5d+147) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = sin(y) * ((x * x) * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9.6) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / Math.sin(y));
	} else if (x <= 1.1e+64) {
		tmp = Math.cosh(x);
	} else if (x <= 7.5e+147) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * ((x * x) * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9.6:
		tmp = (1.0 + (0.5 * (x * x))) / (y / math.sin(y))
	elif x <= 1.1e+64:
		tmp = math.cosh(x)
	elif x <= 7.5e+147:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * ((x * x) * (0.5 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.6)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / Float64(y / sin(y)));
	elseif (x <= 1.1e+64)
		tmp = cosh(x);
	elseif (x <= 7.5e+147)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sin(y) * Float64(Float64(x * x) * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9.6)
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	elseif (x <= 1.1e+64)
		tmp = cosh(x);
	elseif (x <= 7.5e+147)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * ((x * x) * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.6], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+64], N[Cosh[x], $MachinePrecision], If[LessEqual[x, 7.5e+147], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.59999999999999964

   1. Initial program 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 85.8%

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

      1. unpow2 85.8%

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

   8. Simplified 85.8%

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

   9. Taylor expanded in y around inf 82.1%

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

       1. associate-/l* 82.1%

          \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{\frac{y}{\sin y}}} \]

       2. unpow2 82.1%

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

   11. Simplified 82.1%

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

   if 9.59999999999999964 < x < 1.10000000000000001e64

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 86.4%

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

   if 1.10000000000000001e64 < x < 7.50000000000000037e147

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 87.5%

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

      1. unpow2 33.0%

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

   4. Simplified 87.5%

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

   if 7.50000000000000037e147 < x

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. unpow2 100.0%

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

       2. associate-*l/ 71.3%

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

       3. *-commutative 71.3%

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

       4. associate-*l/ 100.0%

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

       5. unpow2 100.0%

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

       6. *-lft-identity 100.0%

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

       7. associate-*l/ 100.0%

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

       8. *-commutative 100.0%

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

       9. associate-*r* 100.0%

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

       10. unpow2 100.0%

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

       11. associate-*l/ 100.0%

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

       12. metadata-eval 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\cosh x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+147}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\left(x \cdot x\right) \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 5: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 64.8%

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

3. Final simplification 64.8%

   \[\leadsto \cosh x \]

Alternative 6: 45.7% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+239}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+239)
   (+ 1.0 (* 0.5 (* x x)))
   (+ 1.0 (* -0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+239) {
		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 <= 5.5d+239) 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 <= 5.5e+239) {
		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 <= 5.5e+239:
		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 <= 5.5e+239)
		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 <= 5.5e+239)
		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, 5.5e+239], 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 < 5.5000000000000004e239

   1. Initial program 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 80.8%

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

      1. unpow2 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in y around 0 49.9%

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

       1. unpow2 49.9%

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

   11. Simplified 49.9%

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

   if 5.5000000000000004e239 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 39.6%

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

   3. Taylor expanded in y around 0 39.5%

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

      1. unpow2 39.5%

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

   5. Simplified 39.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+239}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 7: 32.0% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 49.2%

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

3. Taylor expanded in y around 0 32.9%

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

   1. unpow2 32.9%

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

5. Simplified 32.9%

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

6. Final simplification 32.9%

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

Alternative 8: 26.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 49.2%

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

3. Taylor expanded in y around 0 26.5%

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

4. Final simplification 26.5%

   \[\leadsto 1 \]

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 2023229 
(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)))