2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.5%

Time: 8.4s

Alternatives: 11

Speedup: 76.4×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := 0.5 - t\_0\\ t_2 := 0.5 + t\_0\\ t_3 := \frac{t\_1}{t\_2}\\ t_4 := 1 + t\_3\\ t_5 := \mathsf{fma}\left(-0.5, t\_4, 0.16666666666666666 \cdot t\_3\right) - \frac{t\_4 \cdot t\_1}{t\_2}\\ t_6 := t\_4 \cdot \sin x\\ t_7 := \frac{t\_6}{\cos x}\\ \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(t\_7, -0.5, \frac{\mathsf{fma}\left(t\_5 + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot t\_6\right)}{\cos x}\right) - 0.16666666666666666\right) - t\_5\right) \cdot \varepsilon - \left(-t\_7\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 x))))
        (t_1 (- 0.5 t_0))
        (t_2 (+ 0.5 t_0))
        (t_3 (/ t_1 t_2))
        (t_4 (+ 1.0 t_3))
        (t_5
         (- (fma -0.5 t_4 (* 0.16666666666666666 t_3)) (/ (* t_4 t_1) t_2)))
        (t_6 (* t_4 (sin x)))
        (t_7 (/ t_6 (cos x))))
   (*
    (-
     (fma
      (-
       (*
        (-
         (-
          (*
           (- eps)
           (fma
            t_7
            -0.5
            (/
             (fma
              (+ t_5 0.16666666666666666)
              (sin x)
              (* 0.16666666666666666 t_6))
             (cos x))))
          0.16666666666666666)
         t_5)
        eps)
       (- t_7))
      eps
      1.0)
     (- (pow (* (sin x) (/ 1.0 (cos x))) 2.0)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = 0.5 * cos((2.0 * x));
	double t_1 = 0.5 - t_0;
	double t_2 = 0.5 + t_0;
	double t_3 = t_1 / t_2;
	double t_4 = 1.0 + t_3;
	double t_5 = fma(-0.5, t_4, (0.16666666666666666 * t_3)) - ((t_4 * t_1) / t_2);
	double t_6 = t_4 * sin(x);
	double t_7 = t_6 / cos(x);
	return (fma((((((-eps * fma(t_7, -0.5, (fma((t_5 + 0.16666666666666666), sin(x), (0.16666666666666666 * t_6)) / cos(x)))) - 0.16666666666666666) - t_5) * eps) - -t_7), eps, 1.0) - -pow((sin(x) * (1.0 / cos(x))), 2.0)) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(0.5 * cos(Float64(2.0 * x)))
	t_1 = Float64(0.5 - t_0)
	t_2 = Float64(0.5 + t_0)
	t_3 = Float64(t_1 / t_2)
	t_4 = Float64(1.0 + t_3)
	t_5 = Float64(fma(-0.5, t_4, Float64(0.16666666666666666 * t_3)) - Float64(Float64(t_4 * t_1) / t_2))
	t_6 = Float64(t_4 * sin(x))
	t_7 = Float64(t_6 / cos(x))
	return Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(-eps) * fma(t_7, -0.5, Float64(fma(Float64(t_5 + 0.16666666666666666), sin(x), Float64(0.16666666666666666 * t_6)) / cos(x)))) - 0.16666666666666666) - t_5) * eps) - Float64(-t_7)), eps, 1.0) - Float64(-(Float64(sin(x) * Float64(1.0 / cos(x))) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.5 * t$95$4 + N[(0.16666666666666666 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$4 * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$7 * -0.5 + N[(N[(N[(t$95$5 + 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - t$95$5), $MachinePrecision] * eps), $MachinePrecision] - (-t$95$7)), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[Power[N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]]]]]]]]]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]
5. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. quot-tan N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. mult-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   5. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   7. lift-cos.f64 99.5

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

6. Applied rewrites 99.5%

   \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. quot-tan N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. mult-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   5. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   7. lift-cos.f64 99.5

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 99.5%

   \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. quot-tan N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. mult-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   5. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   7. lift-cos.f64 99.5

      \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

10. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
11. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

14. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
15. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

16. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
17. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

18. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
19. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

20. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
21. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

22. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
23. Step-by-step derivation

    1. lift-tan.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. quot-tan N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. mult-flip N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. lift-sin.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 99.5

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

24. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

25. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
26. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

27. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

28. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
29. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

30. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

31. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
32. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

33. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

34. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
35. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

36. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right), -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

37. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
38. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

39. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

40. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
41. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

42. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

43. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
44. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\sin x \cdot \sin x}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    3. sqr-sin-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    5. sqr-cos-a-rev N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-/.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-cos.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    10. lift--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-cos.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

    14. lift-+.f64 99.5

        \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

45. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

46. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
47. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\mathsf{fma}\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{-1}{2}, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

48. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.5, 1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, 0.16666666666666666 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) - \frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, -0.5, {\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

49. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{2}, 1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{1}{6} \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) - \frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\left(\frac{-1}{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
50. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, \frac{-1}{2}, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{2}, 1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}, \frac{1}{6} \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) - \frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) + \frac{1}{6}, \sin x, \frac{1}{6} \cdot \left(\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - \frac{1}{6}\right) - \left(\left(\frac{-1}{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]

51. Applied rewrites 99.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.5, 1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, 0.16666666666666666 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) - \frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}, 0.16666666666666666 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) - \frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\left(\sin x \cdot \frac{1}{\cos x}\right)}^{2}\right)\right) \cdot \varepsilon \]
52. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := {\tan x}^{2}\\ t_2 := 1 + t\_1\\ t_3 := \mathsf{fma}\left(t\_2, -0.5, t\_1 \cdot 0.16666666666666666\right) - \left(0.5 - 0.5 \cdot t\_0\right) \cdot \frac{t\_2}{0.5 - -0.5 \cdot t\_0}\\ t_4 := t\_2 \cdot \sin x\\ t_5 := \frac{t\_4}{\cos x}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \mathsf{fma}\left(t\_5, -0.5, \frac{\mathsf{fma}\left(t\_3 - -0.16666666666666666, \sin x, 0.16666666666666666 \cdot t\_4\right)}{\cos x}\right), -0.16666666666666666\right) - t\_3, \varepsilon, t\_5\right), \varepsilon, 1\right) + t\_1\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x)))
        (t_1 (pow (tan x) 2.0))
        (t_2 (+ 1.0 t_1))
        (t_3
         (-
          (fma t_2 -0.5 (* t_1 0.16666666666666666))
          (* (- 0.5 (* 0.5 t_0)) (/ t_2 (- 0.5 (* -0.5 t_0))))))
        (t_4 (* t_2 (sin x)))
        (t_5 (/ t_4 (cos x))))
   (*
    (+
     (fma
      (fma
       (-
        (fma
         (- eps)
         (fma
          t_5
          -0.5
          (/
           (fma
            (- t_3 -0.16666666666666666)
            (sin x)
            (* 0.16666666666666666 t_4))
           (cos x)))
         -0.16666666666666666)
        t_3)
       eps
       t_5)
      eps
      1.0)
     t_1)
    eps)))

C

double code(double x, double eps) {
	double t_0 = cos((2.0 * x));
	double t_1 = pow(tan(x), 2.0);
	double t_2 = 1.0 + t_1;
	double t_3 = fma(t_2, -0.5, (t_1 * 0.16666666666666666)) - ((0.5 - (0.5 * t_0)) * (t_2 / (0.5 - (-0.5 * t_0))));
	double t_4 = t_2 * sin(x);
	double t_5 = t_4 / cos(x);
	return (fma(fma((fma(-eps, fma(t_5, -0.5, (fma((t_3 - -0.16666666666666666), sin(x), (0.16666666666666666 * t_4)) / cos(x))), -0.16666666666666666) - t_3), eps, t_5), eps, 1.0) + t_1) * eps;
}

Julia

function code(x, eps)
	t_0 = cos(Float64(2.0 * x))
	t_1 = tan(x) ^ 2.0
	t_2 = Float64(1.0 + t_1)
	t_3 = Float64(fma(t_2, -0.5, Float64(t_1 * 0.16666666666666666)) - Float64(Float64(0.5 - Float64(0.5 * t_0)) * Float64(t_2 / Float64(0.5 - Float64(-0.5 * t_0)))))
	t_4 = Float64(t_2 * sin(x))
	t_5 = Float64(t_4 / cos(x))
	return Float64(Float64(fma(fma(Float64(fma(Float64(-eps), fma(t_5, -0.5, Float64(fma(Float64(t_3 - -0.16666666666666666), sin(x), Float64(0.16666666666666666 * t_4)) / cos(x))), -0.16666666666666666) - t_3), eps, t_5), eps, 1.0) + t_1) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * -0.5 + N[(t$95$1 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(0.5 - N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$5 * -0.5 + N[(N[(N[(t$95$3 - -0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] - t$95$3), $MachinePrecision] * eps + t$95$5), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] * eps), $MachinePrecision]]]]]]]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \mathsf{fma}\left(\frac{\left(1 + {\tan x}^{2}\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 + {\tan x}^{2}, -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + {\tan x}^{2}}{0.5 - -0.5 \cdot \cos \left(2 \cdot x\right)}\right) - -0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 + {\tan x}^{2}\right) \cdot \sin x\right)\right)}{\cos x}\right), -0.16666666666666666\right) - \left(\mathsf{fma}\left(1 + {\tan x}^{2}, -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 + {\tan x}^{2}}{0.5 - -0.5 \cdot \cos \left(2 \cdot x\right)}\right), \varepsilon, \frac{\left(1 + {\tan x}^{2}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon} \]
7. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := {\tan x}^{2}\\ t_2 := -t\_1\\ t_3 := 1 - t\_2\\ \left(\mathsf{fma}\left(\left(-0.16666666666666666 - \left(\mathsf{fma}\left(t\_3, -0.5, t\_1 \cdot 0.16666666666666666\right) + \left(-\left(0.5 - t\_0\right) \cdot \frac{t\_3}{0.5 + t\_0}\right)\right)\right) \cdot \varepsilon - \left(-\frac{t\_3 \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - t\_2\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 x))))
        (t_1 (pow (tan x) 2.0))
        (t_2 (- t_1))
        (t_3 (- 1.0 t_2)))
   (*
    (-
     (fma
      (-
       (*
        (-
         -0.16666666666666666
         (+
          (fma t_3 -0.5 (* t_1 0.16666666666666666))
          (- (* (- 0.5 t_0) (/ t_3 (+ 0.5 t_0))))))
        eps)
       (- (/ (* t_3 (sin x)) (cos x))))
      eps
      1.0)
     t_2)
    eps)))

C

double code(double x, double eps) {
	double t_0 = 0.5 * cos((2.0 * x));
	double t_1 = pow(tan(x), 2.0);
	double t_2 = -t_1;
	double t_3 = 1.0 - t_2;
	return (fma((((-0.16666666666666666 - (fma(t_3, -0.5, (t_1 * 0.16666666666666666)) + -((0.5 - t_0) * (t_3 / (0.5 + t_0))))) * eps) - -((t_3 * sin(x)) / cos(x))), eps, 1.0) - t_2) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(0.5 * cos(Float64(2.0 * x)))
	t_1 = tan(x) ^ 2.0
	t_2 = Float64(-t_1)
	t_3 = Float64(1.0 - t_2)
	return Float64(Float64(fma(Float64(Float64(Float64(-0.16666666666666666 - Float64(fma(t_3, -0.5, Float64(t_1 * 0.16666666666666666)) + Float64(-Float64(Float64(0.5 - t_0) * Float64(t_3 / Float64(0.5 + t_0)))))) * eps) - Float64(-Float64(Float64(t_3 * sin(x)) / cos(x)))), eps, 1.0) - t_2) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(1.0 - t$95$2), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(-0.16666666666666666 - N[(N[(t$95$3 * -0.5 + N[(t$95$1 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(t$95$3 / N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] - (-N[(N[(t$95$3 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps + 1.0), $MachinePrecision] - t$95$2), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{6} - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \frac{-1}{2}, {\tan x}^{2} \cdot \frac{1}{6}\right) + \left(-\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(\left(-0.16666666666666666 - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -{\tan x}^{2}\\ \left(\mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon - \left(-\frac{\left(1 - t\_0\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (tan x) 2.0))))
   (*
    (-
     (fma
      (- (* 0.3333333333333333 eps) (- (/ (* (- 1.0 t_0) (sin x)) (cos x))))
      eps
      1.0)
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = -pow(tan(x), 2.0);
	return (fma(((0.3333333333333333 * eps) - -(((1.0 - t_0) * sin(x)) / cos(x))), eps, 1.0) - t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(-(tan(x) ^ 2.0))
	return Float64(Float64(fma(Float64(Float64(0.3333333333333333 * eps) - Float64(-Float64(Float64(Float64(1.0 - t_0) * sin(x)) / cos(x)))), eps, 1.0) - t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])}, N[(N[(N[(N[(N[(0.3333333333333333 * eps), $MachinePrecision] - (-N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - 0.16666666666666666\right) - \left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + \left(-\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1 - \left(-{\tan x}^{2}\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right)\right)\right) \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

7. Applied rewrites 99.3%

   \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon - \left(-\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 + t\_0\right) \cdot \sin x}{\cos x}, 1\right) + t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (* (+ (fma eps (/ (* (+ 1.0 t_0) (sin x)) (cos x)) 1.0) t_0) eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (fma(eps, (((1.0 + t_0) * sin(x)) / cos(x)), 1.0) + t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(fma(eps, Float64(Float64(Float64(1.0 + t_0) * sin(x)) / cos(x)), 1.0) + t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(eps * N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]
5. Step-by-step derivation

6. Applied rewrites 99.2%

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 + {\tan x}^{2}\right) \cdot \sin x}{\cos x}, 1\right) + {\tan x}^{2}\right) \cdot \color{blue}{\varepsilon} \]
7. Add Preprocessing

Alternative 6: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\varepsilon, \frac{1 \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (- (fma eps (/ (* 1.0 (sin x)) (cos x)) 1.0) (- (pow (tan x) 2.0))) eps))

C

double code(double x, double eps) {
	return (fma(eps, ((1.0 * sin(x)) / cos(x)), 1.0) - -pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(eps, Float64(Float64(1.0 * sin(x)) / cos(x)), 1.0) - Float64(-(tan(x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * N[(N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \frac{1 \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \frac{1 \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 7: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\varepsilon, x, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (- (fma eps x 1.0) (- (pow (tan x) 2.0))) eps))

C

double code(double x, double eps) {
	return (fma(eps, x, 1.0) - -pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(eps, x, 1.0) - Float64(-(tan(x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * x + 1.0), $MachinePrecision] - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, x, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, x, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 8: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (- 1.0 (- (pow (tan x) 2.0))) eps))

C

double code(double x, double eps) {
	return (1.0 - -pow(tan(x), 2.0)) * eps;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (1.0d0 - -(tan(x) ** 2.0d0)) * eps
end function

Java

public static double code(double x, double eps) {
	return (1.0 - -Math.pow(Math.tan(x), 2.0)) * eps;
}

Python

def code(x, eps):
	return (1.0 - -math.pow(math.tan(x), 2.0)) * eps

Julia

function code(x, eps)
	return Float64(Float64(1.0 - Float64(-(tan(x) ^ 2.0))) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = (1.0 - -(tan(x) ^ 2.0)) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(1.0 - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   3. lower--.f64 N/A

      \[\leadsto \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon \]

   4. mul-1-neg N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \cdot \varepsilon \]

   5. unpow2 N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\frac{\sin x \cdot \sin x}{{\cos x}^{2}}\right)\right)\right) \cdot \varepsilon \]

   6. unpow2 N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\frac{\sin x \cdot \sin x}{\cos x \cdot \cos x}\right)\right)\right) \cdot \varepsilon \]

   7. frac-times N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)\right) \cdot \varepsilon \]

   8. quot-tan N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\tan x \cdot \frac{\sin x}{\cos x}\right)\right)\right) \cdot \varepsilon \]

   9. quot-tan N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)\right) \cdot \varepsilon \]

   10. lower-neg.f64 N/A

       \[\leadsto \left(1 - \left(-\tan x \cdot \tan x\right)\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   13. lift-tan.f64 98.8

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon + x \cdot \mathsf{fma}\left(\varepsilon, x, \varepsilon \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* x (fma eps x (* eps eps)))))

C

double code(double x, double eps) {
	return eps + (x * fma(eps, x, (eps * eps)));
}

Julia

function code(x, eps)
	return Float64(eps + Float64(x * fma(eps, x, Float64(eps * eps))))
end

Wolfram

code[x_, eps_] := N[(eps + N[(x * N[(eps * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \varepsilon + x \cdot \color{blue}{\left(\varepsilon \cdot x + {\varepsilon}^{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \varepsilon + x \cdot \left(\varepsilon \cdot x + \color{blue}{{\varepsilon}^{2}}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \varepsilon + x \cdot \mathsf{fma}\left(\varepsilon, x, {\varepsilon}^{2}\right) \]

   4. unpow2 N/A

      \[\leadsto \varepsilon + x \cdot \mathsf{fma}\left(\varepsilon, x, \varepsilon \cdot \varepsilon\right) \]

   5. lower-*.f64 98.1

      \[\leadsto \varepsilon + x \cdot \mathsf{fma}\left(\varepsilon, x, \varepsilon \cdot \varepsilon\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \varepsilon + \color{blue}{x \cdot \mathsf{fma}\left(\varepsilon, x, \varepsilon \cdot \varepsilon\right)} \]
8. Add Preprocessing

Alternative 10: 98.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (+ 1.0 (* x (+ eps x))) eps))

C

double code(double x, double eps) {
	return (1.0 + (x * (eps + x))) * eps;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (1.0d0 + (x * (eps + x))) * eps
end function

Java

public static double code(double x, double eps) {
	return (1.0 + (x * (eps + x))) * eps;
}

Python

def code(x, eps):
	return (1.0 + (x * (eps + x))) * eps

Julia

function code(x, eps)
	return Float64(Float64(1.0 + Float64(x * Float64(eps + x))) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = (1.0 + (x * (eps + x))) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(1.0 + N[(x * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]

   3. lower-+.f64 98.1

      \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]

7. Applied rewrites 98.1%

   \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 11: 97.6% accurate, 76.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\varepsilon, \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \varepsilon \cdot x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(1 + \varepsilon \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 97.6

      \[\leadsto \left(1 + \varepsilon \cdot x\right) \cdot \varepsilon \]

7. Applied rewrites 97.6%

   \[\leadsto \left(1 + \varepsilon \cdot x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 97.6%

    \[\leadsto \varepsilon \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Developer Target 2: 62.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))

C

double code(double x, double eps) {
	return ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
}

Python

def code(x, eps):
	return ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)

Julia

function code(x, eps)
	return Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Developer Target 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \left(\varepsilon \cdot \tan x\right) \cdot \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* (* eps (tan x)) (tan x))))

C

double code(double x, double eps) {
	return eps + ((eps * tan(x)) * tan(x));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((eps * tan(x)) * tan(x))
end function

Java

public static double code(double x, double eps) {
	return eps + ((eps * Math.tan(x)) * Math.tan(x));
}

Python

def code(x, eps):
	return eps + ((eps * math.tan(x)) * math.tan(x))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(eps * tan(x)) * tan(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((eps * tan(x)) * tan(x));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(eps * N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025142 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform c (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform c (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform c (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))