Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.2% → 99.7%

Time: 11.9s

Alternatives: 17

Speedup: 3.4×

• Could not converge on a ground truth

  1. x = 3.112407680001184e+120
  2. y = 1.6732628018052553e+285
  3. z = 3.0654901728601654e+28

• 63.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\cosh x\_m}{\frac{x\_m}{y\_m} \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x\_m}{z\_m} \cdot y\_m}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 5e+94)
      (/ (cosh x_m) (* (/ x_m y_m) z_m))
      (/ (* (/ (cosh x_m) z_m) y_m) x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5e+94) {
		tmp = cosh(x_m) / ((x_m / y_m) * z_m);
	} else {
		tmp = ((cosh(x_m) / z_m) * y_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5d+94) then
        tmp = cosh(x_m) / ((x_m / y_m) * z_m)
    else
        tmp = ((cosh(x_m) / z_m) * y_m) / x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * Math.cosh(x_m)) / z_m) <= 5e+94) {
		tmp = Math.cosh(x_m) / ((x_m / y_m) * z_m);
	} else {
		tmp = ((Math.cosh(x_m) / z_m) * y_m) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, y_s, z_s, x_m, y_m, z_m):
	tmp = 0
	if (((y_m / x_m) * math.cosh(x_m)) / z_m) <= 5e+94:
		tmp = math.cosh(x_m) / ((x_m / y_m) * z_m)
	else:
		tmp = ((math.cosh(x_m) / z_m) * y_m) / x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 5e+94)
		tmp = Float64(cosh(x_m) / Float64(Float64(x_m / y_m) * z_m));
	else
		tmp = Float64(Float64(Float64(cosh(x_m) / z_m) * y_m) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 5e+94)
		tmp = cosh(x_m) / ((x_m / y_m) * z_m);
	else
		tmp = ((cosh(x_m) / z_m) * y_m) / x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 5e+94], N[(N[Cosh[x$95$m], $MachinePrecision] / N[(N[(x$95$m / y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cosh[x$95$m], $MachinePrecision] / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5.0000000000000001e94

   1. Initial program 99.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 5.0000000000000001e94 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 79.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\cosh x}{\frac{x}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{z} \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{y\_m}{x\_m} \cdot \cosh x\_m}{z\_m} \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{y\_m}{\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, 0.041666666666666664\right), 0.5\right) \cdot x\_m, x\_m, 1\right)}} \cdot \frac{\frac{-1}{x\_m}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, 0.041666666666666664\right) \cdot x\_m, x\_m, 0.5\right), x\_m \cdot x\_m, 1\right)}}}{x\_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s y_s z_s x_m y_m z_m)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* (/ y_m x_m) (cosh x_m)) z_m) 4e-51)
      (*
       (/
        y_m
        (/
         -1.0
         (fma
          (*
           (fma
            (* x_m x_m)
            (fma (* x_m x_m) 0.001388888888888889 0.041666666666666664)
            0.5)
           x_m)
          x_m
          1.0)))
       (/ (/ -1.0 x_m) z_m))
      (/
       (/
        y_m
        (/
         z_m
         (fma
          (fma
           (* (fma (* 0.001388888888888889 x_m) x_m 0.041666666666666664) x_m)
           x_m
           0.5)
          (* x_m x_m)
          1.0)))
       x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((((y_m / x_m) * cosh(x_m)) / z_m) <= 4e-51) {
		tmp = (y_m / (-1.0 / fma((fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5) * x_m), x_m, 1.0))) * ((-1.0 / x_m) / z_m);
	} else {
		tmp = (y_m / (z_m / fma(fma((fma((0.001388888888888889 * x_m), x_m, 0.041666666666666664) * x_m), x_m, 0.5), (x_m * x_m), 1.0))) / x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, y_s, z_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m / x_m) * cosh(x_m)) / z_m) <= 4e-51)
		tmp = Float64(Float64(y_m / Float64(-1.0 / fma(Float64(fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, 0.041666666666666664), 0.5) * x_m), x_m, 1.0))) * Float64(Float64(-1.0 / x_m) / z_m));
	else
		tmp = Float64(Float64(y_m / Float64(z_m / fma(fma(Float64(fma(Float64(0.001388888888888889 * x_m), x_m, 0.041666666666666664) * x_m), x_m, 0.5), Float64(x_m * x_m), 1.0))) / x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 4e-51], N[(N[(y$95$m / N[(-1.0 / N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z$95$m / N[(N[(N[(N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + 0.041666666666666664), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4e-51

   1. Initial program 99.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 71.7

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1}{z}}{x} \]

      3. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} + 1}{z}}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, x, 1\right)}}{z}}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x}, x, 1\right)}{z}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x, x, 1\right)}{z}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)} \cdot x, x, 1\right)}{z}}{x} \]

      10. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      14. unpow2 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      15. lower-*.f64 70.7

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \cdot x, x, 1\right)}{z}}{x} \]

   7. Applied rewrites 70.7%

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}{z}}{x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}}{x} \]

      3. clear-num N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      6. lower-/.f64 70.7

         \[\leadsto \frac{\frac{y}{\color{blue}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}}}{x} \]

   9. Applied rewrites 70.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right)}}}}{x} \]

   11. Applied rewrites 98.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{-z} \cdot \frac{y}{\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right) \cdot x, x, 1\right)}}} \]

   if 4e-51 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 82.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}}{z}}{x} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1}{z}}{x} \]

      3. unpow2 N/A

         \[\leadsto \frac{y \cdot \frac{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} + 1}{z}}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x, x, 1\right)}}{z}}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot x}, x, 1\right)}{z}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \cdot x, x, 1\right)}{z}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)} \cdot x, x, 1\right)}{z}}{x} \]

      10. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      12. unpow2 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      14. unpow2 N/A

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}{x} \]

      15. lower-*.f64 93.7

          \[\leadsto \frac{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \cdot x, x, 1\right)}{z}}{x} \]

   7. Applied rewrites 93.7%

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}{z}}{x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}}{x} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}{z}}}{x} \]

      3. clear-num N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right) \cdot x, x, 1\right)}}}}{x} \]

      6. lower-/.f64 93.6

         \[\leadsto \frac{\frac{y}{\color{blue}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}}}}{x} \]

   9. Applied rewrites 93.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right)}}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{x} \cdot \cosh x}{z} \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right) \cdot x, x, 1\right)}} \cdot \frac{\frac{-1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot x, x, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right)}}}{x}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))