Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.8%

Time: 3.6s

Alternatives: 5

Speedup: N/A×

• 63.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 84.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x\_m \cdot \frac{y\_m}{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \cosh x\_m}{x\_m}}{z} \cdot y\_m}{2}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 5e+275)
       (/ t_0 z)
       (/ (* (/ (/ (* 2.0 (cosh x_m)) x_m) z) y_m) 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 5e+275) {
		tmp = t_0 / z;
	} else {
		tmp = ((((2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m =     private
x\_s =     private
y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 5d+275) then
        tmp = t_0 / z
    else
        tmp = ((((2.0d0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0d0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 5e+275) {
		tmp = t_0 / z;
	} else {
		tmp = ((((2.0 * Math.cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 5e+275:
		tmp = t_0 / z
	else:
		tmp = ((((2.0 * math.cosh(x_m)) / x_m) / z) * y_m) / 2.0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 5e+275)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 5e+275)
		tmp = t_0 / z;
	else
		tmp = ((((2.0 * cosh(x_m)) / x_m) / z) * y_m) / 2.0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+275], N[(t$95$0 / z), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000003e275

   1. Initial program 99.7%

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

   if 5.0000000000000003e275 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. 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. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. cosh-def N/A

         \[\leadsto \color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot \frac{\frac{y}{x}}{z} \]

      7. rec-exp N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 2: 94.2% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(2 \cdot \cosh x\_m\right) \cdot y\_m}{2 \cdot \left(z \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, \frac{\left(x\_m \cdot x\_m\right) \cdot y\_m}{z}, 0.041666666666666664 \cdot \frac{y\_m}{z}\right) \cdot x\_m, x\_m, \frac{y\_m}{z} \cdot 0.5\right), \frac{x\_m \cdot x\_m}{x\_m}, \frac{\frac{y\_m}{z}}{x\_m}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 1.1e+93)
     (/ (* (* 2.0 (cosh x_m)) y_m) (* 2.0 (* z x_m)))
     (fma
      (fma
       (*
        (fma
         0.001388888888888889
         (/ (* (* x_m x_m) y_m) z)
         (* 0.041666666666666664 (/ y_m z)))
        x_m)
       x_m
       (* (/ y_m z) 0.5))
      (/ (* x_m x_m) x_m)
      (/ (/ y_m z) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.1e+93) {
		tmp = ((2.0 * cosh(x_m)) * y_m) / (2.0 * (z * x_m));
	} else {
		tmp = fma(fma((fma(0.001388888888888889, (((x_m * x_m) * y_m) / z), (0.041666666666666664 * (y_m / z))) * x_m), x_m, ((y_m / z) * 0.5)), ((x_m * x_m) / x_m), ((y_m / z) / x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.1e+93)
		tmp = Float64(Float64(Float64(2.0 * cosh(x_m)) * y_m) / Float64(2.0 * Float64(z * x_m)));
	else
		tmp = fma(fma(Float64(fma(0.001388888888888889, Float64(Float64(Float64(x_m * x_m) * y_m) / z), Float64(0.041666666666666664 * Float64(y_m / z))) * x_m), x_m, Float64(Float64(y_m / z) * 0.5)), Float64(Float64(x_m * x_m) / x_m), Float64(Float64(y_m / z) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.1e+93], N[(N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(2.0 * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] + N[(0.041666666666666664 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(y$95$m / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000011e93

   1. Initial program 91.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. 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. lift-cosh.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. cosh-def N/A

         \[\leadsto \color{blue}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{2}} \cdot \frac{\frac{y}{x}}{z} \]

      7. rec-exp N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. rec-exp N/A

          \[\leadsto \frac{\left(e^{x} + \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \cdot y}{2 \cdot \left(x \cdot z\right)} \]

      15. cosh-undef N/A

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

      16. lower-*.f64 N/A

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

      17. lift-cosh.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 92.9

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

   3. Applied rewrites 92.9%

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

   if 1.10000000000000011e93 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in x around 0

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

      1. div-add N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 96.0%

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

Alternative 3: 94.0% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\cosh x\_m \cdot \frac{y\_m}{x\_m}}{z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, \frac{\left(x\_m \cdot x\_m\right) \cdot y\_m}{z}, 0.041666666666666664 \cdot \frac{y\_m}{z}\right) \cdot x\_m, x\_m, \frac{y\_m}{z} \cdot 0.5\right), \frac{x\_m \cdot x\_m}{x\_m}, \frac{\frac{y\_m}{z}}{x\_m}\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x_m) (/ y_m x_m)) z)))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 INFINITY)
       t_0
       (fma
        (fma
         (*
          (fma
           0.001388888888888889
           (/ (* (* x_m x_m) y_m) z)
           (* 0.041666666666666664 (/ y_m z)))
          x_m)
         x_m
         (* (/ y_m z) 0.5))
        (/ (* x_m x_m) x_m)
        (/ (/ y_m z) x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (cosh(x_m) * (y_m / x_m)) / z;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(fma((fma(0.001388888888888889, (((x_m * x_m) * y_m) / z), (0.041666666666666664 * (y_m / z))) * x_m), x_m, ((y_m / z) * 0.5)), ((x_m * x_m) / x_m), ((y_m / z) / x_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(cosh(x_m) * Float64(y_m / x_m)) / z)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = fma(fma(Float64(fma(0.001388888888888889, Float64(Float64(Float64(x_m * x_m) * y_m) / z), Float64(0.041666666666666664 * Float64(y_m / z))) * x_m), x_m, Float64(Float64(y_m / z) * 0.5)), Float64(Float64(x_m * x_m) / x_m), Float64(Float64(y_m / z) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(0.001388888888888889 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] + N[(0.041666666666666664 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(y$95$m / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(y$95$m / z), $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) < +inf.0

   1. Initial program 95.1%

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

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. div-add N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 87.0%

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

Alternative 4: 83.6% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, \frac{\left(x\_m \cdot x\_m\right) \cdot y\_m}{z}, 0.041666666666666664 \cdot \frac{y\_m}{z}\right) \cdot x\_m, x\_m, \frac{y\_m}{z} \cdot 0.5\right), \frac{x\_m \cdot x\_m}{x\_m}, \frac{\frac{y\_m}{z}}{x\_m}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (fma
    (fma
     (*
      (fma
       0.001388888888888889
       (/ (* (* x_m x_m) y_m) z)
       (* 0.041666666666666664 (/ y_m z)))
      x_m)
     x_m
     (* (/ y_m z) 0.5))
    (/ (* x_m x_m) x_m)
    (/ (/ y_m z) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * fma(fma((fma(0.001388888888888889, (((x_m * x_m) * y_m) / z), (0.041666666666666664 * (y_m / z))) * x_m), x_m, ((y_m / z) * 0.5)), ((x_m * x_m) / x_m), ((y_m / z) / x_m)));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * fma(fma(Float64(fma(0.001388888888888889, Float64(Float64(Float64(x_m * x_m) * y_m) / z), Float64(0.041666666666666664 * Float64(y_m / z))) * x_m), x_m, Float64(Float64(y_m / z) * 0.5)), Float64(Float64(x_m * x_m) / x_m), Float64(Float64(y_m / z) / x_m))))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(N[(0.001388888888888889 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision] + N[(0.041666666666666664 * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(N[(y$95$m / z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in x around 0

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

   1. div-add N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 83.6%

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

Alternative 5: 45.3% accurate, N/A× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \left(\mathsf{fma}\left(\frac{y\_m}{z}, 0.001388888888888889, \mathsf{fma}\left(\frac{y\_m}{\left(x\_m \cdot x\_m\right) \cdot z}, 0.041666666666666664, \mathsf{fma}\left(\frac{y\_m}{{\left(x\_m \cdot x\_m\right)}^{2} \cdot z}, 0.5, \frac{y\_m}{{\left(x\_m \cdot x\_m\right)}^{3} \cdot z}\right)\right)\right) \cdot {\left(x\_m \cdot x\_m\right)}^{2.5}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (*
    (fma
     (/ y_m z)
     0.001388888888888889
     (fma
      (/ y_m (* (* x_m x_m) z))
      0.041666666666666664
      (fma
       (/ y_m (* (pow (* x_m x_m) 2.0) z))
       0.5
       (/ y_m (* (pow (* x_m x_m) 3.0) z)))))
    (pow (* x_m x_m) 2.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (fma((y_m / z), 0.001388888888888889, fma((y_m / ((x_m * x_m) * z)), 0.041666666666666664, fma((y_m / (pow((x_m * x_m), 2.0) * z)), 0.5, (y_m / (pow((x_m * x_m), 3.0) * z))))) * pow((x_m * x_m), 2.5)));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(fma(Float64(y_m / z), 0.001388888888888889, fma(Float64(y_m / Float64(Float64(x_m * x_m) * z)), 0.041666666666666664, fma(Float64(y_m / Float64((Float64(x_m * x_m) ^ 2.0) * z)), 0.5, Float64(y_m / Float64((Float64(x_m * x_m) ^ 3.0) * z))))) * (Float64(x_m * x_m) ^ 2.5))))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(y$95$m / z), $MachinePrecision] * 0.001388888888888889 + N[(N[(y$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(N[(y$95$m / N[(N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(y$95$m / N[(N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 3.0], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(x$95$m * x$95$m), $MachinePrecision], 2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in x around 0

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

   1. div-add N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 83.6%

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

5. Taylor expanded in x around inf

   \[\leadsto {x}^{5} \cdot \color{blue}{\left(\frac{1}{720} \cdot \frac{y}{z} + \left(\frac{1}{24} \cdot \frac{y}{{x}^{2} \cdot z} + \left(\frac{1}{2} \cdot \frac{y}{{x}^{4} \cdot z} + \frac{y}{{x}^{6} \cdot z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{720} \cdot \frac{y}{z} + \left(\frac{1}{24} \cdot \frac{y}{{x}^{2} \cdot z} + \left(\frac{1}{2} \cdot \frac{y}{{x}^{4} \cdot z} + \frac{y}{{x}^{6} \cdot z}\right)\right)\right) \cdot {x}^{\color{blue}{5}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{720} \cdot \frac{y}{z} + \left(\frac{1}{24} \cdot \frac{y}{{x}^{2} \cdot z} + \left(\frac{1}{2} \cdot \frac{y}{{x}^{4} \cdot z} + \frac{y}{{x}^{6} \cdot z}\right)\right)\right) \cdot {x}^{\color{blue}{5}} \]

7. Applied rewrites 45.3%

   \[\leadsto \mathsf{fma}\left(\frac{y}{z}, 0.001388888888888889, \mathsf{fma}\left(\frac{y}{\left(x \cdot x\right) \cdot z}, 0.041666666666666664, \mathsf{fma}\left(\frac{y}{{\left(x \cdot x\right)}^{2} \cdot z}, 0.5, \frac{y}{{\left(x \cdot x\right)}^{3} \cdot z}\right)\right)\right) \cdot \color{blue}{{\left(x \cdot x\right)}^{2.5}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64
  (/ (* (cosh x) (/ y x)) z))