2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.7%

Time: 9.4s

Alternatives: 20

Speedup: 207.0×

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: 62.3% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{3}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + \mathsf{fma}\left(0.3333333333333333, t\_1, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.6666666666666666, \tan x, 0.6666666666666666 \cdot t\_0\right) - -1 \cdot \frac{{\sin x}^{3} \cdot \left(1 + t\_1\right)}{{\cos x}^{3}}, t\_1\right)\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}, \tan x + t\_0\right), t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 3.0)) (t_1 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (fma
       eps
       (-
        (+
         0.3333333333333333
         (fma
          0.3333333333333333
          t_1
          (fma
           eps
           (-
            (fma 0.6666666666666666 (tan x) (* 0.6666666666666666 t_0))
            (* -1.0 (/ (* (pow (sin x) 3.0) (+ 1.0 t_1)) (pow (cos x) 3.0))))
           t_1)))
        (* -1.0 (/ (pow (sin x) 4.0) (pow (cos x) 4.0))))
       (+ (tan x) t_0))
      t_1)))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 3.0);
	double t_1 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (1.0 + fma(eps, fma(eps, ((0.3333333333333333 + fma(0.3333333333333333, t_1, fma(eps, (fma(0.6666666666666666, tan(x), (0.6666666666666666 * t_0)) - (-1.0 * ((pow(sin(x), 3.0) * (1.0 + t_1)) / pow(cos(x), 3.0)))), t_1))) - (-1.0 * (pow(sin(x), 4.0) / pow(cos(x), 4.0)))), (tan(x) + t_0)), t_1));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 3.0
	t_1 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, Float64(Float64(0.3333333333333333 + fma(0.3333333333333333, t_1, fma(eps, Float64(fma(0.6666666666666666, tan(x), Float64(0.6666666666666666 * t_0)) - Float64(-1.0 * Float64(Float64((sin(x) ^ 3.0) * Float64(1.0 + t_1)) / (cos(x) ^ 3.0)))), t_1))) - Float64(-1.0 * Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))), Float64(tan(x) + t_0)), t_1)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(eps * N[(N[(0.3333333333333333 + N[(0.3333333333333333 * t$95$1 + N[(eps * N[(N[(0.6666666666666666 * N[Tan[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(x + \varepsilon\right)} - \tan x \]

   2. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   5. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x + \frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   7. lift-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   8. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   9. lower-tan.f64 N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   11. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   13. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x \]

   14. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   15. lower-tan.f64 62.5

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x \cdot \tan \varepsilon}} - \tan x \]

   3. lift-tan.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \tan \varepsilon} - \tan x \]

   4. lift-tan.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   5. flip3-- N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

   7. metadata-eval N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{\color{blue}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\color{blue}{\tan x} \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   11. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \color{blue}{\tan \varepsilon}\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\color{blue}{\left(\tan x \cdot \tan \varepsilon\right)}}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   13. metadata-eval N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\color{blue}{1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

5. Applied rewrites 62.5%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \tan x \cdot \tan \varepsilon, 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{1}{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\left(\frac{2}{3} \cdot \frac{\sin x}{\cos x} + \frac{2}{3} \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) - -1 \cdot \frac{{\sin x}^{3} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{3}}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + \mathsf{fma}\left(0.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.6666666666666666, \tan x, 0.6666666666666666 \cdot {\tan x}^{3}\right) - -1 \cdot \frac{{\sin x}^{3} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{3}}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}, \tan x + {\tan x}^{3}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := 1 - \left(-t\_0\right)\\ t_2 := t\_1 \cdot \tan x\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.5, \frac{\mathsf{fma}\left(-0.3333333333333333, \sin x, \left(0.16666666666666666 \cdot t\_1\right) \cdot \sin x\right)}{\cos x}\right) \cdot \left(-\varepsilon\right) - \mathsf{fma}\left(t\_1 \cdot t\_3, -1, \mathsf{fma}\left(t\_1, -0.5, t\_0 \cdot 0.16666666666666666\right) + 0.16666666666666666\right), \varepsilon, t\_2\right), \varepsilon, 1 - \left(-t\_3\right)\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (- 1.0 (- t_0)))
        (t_2 (* t_1 (tan x)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    (fma
     (fma
      (-
       (*
        (fma
         t_2
         -0.5
         (/
          (fma
           -0.3333333333333333
           (sin x)
           (* (* 0.16666666666666666 t_1) (sin x)))
          (cos x)))
        (- eps))
       (fma
        (* t_1 t_3)
        -1.0
        (+ (fma t_1 -0.5 (* t_0 0.16666666666666666)) 0.16666666666666666)))
      eps
      t_2)
     eps
     (- 1.0 (- t_3)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = 1.0 - -t_0;
	double t_2 = t_1 * tan(x);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return fma(fma(((fma(t_2, -0.5, (fma(-0.3333333333333333, sin(x), ((0.16666666666666666 * t_1) * sin(x))) / cos(x))) * -eps) - fma((t_1 * t_3), -1.0, (fma(t_1, -0.5, (t_0 * 0.16666666666666666)) + 0.16666666666666666))), eps, t_2), eps, (1.0 - -t_3)) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 - Float64(-t_0))
	t_2 = Float64(t_1 * tan(x))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(fma(fma(Float64(Float64(fma(t_2, -0.5, Float64(fma(-0.3333333333333333, sin(x), Float64(Float64(0.16666666666666666 * t_1) * sin(x))) / cos(x))) * Float64(-eps)) - fma(Float64(t_1 * t_3), -1.0, Float64(fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666)) + 0.16666666666666666))), eps, t_2), eps, Float64(1.0 - Float64(-t_3))) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(t$95$2 * -0.5 + N[(N[(-0.3333333333333333 * N[Sin[x], $MachinePrecision] + N[(N[(0.16666666666666666 * t$95$1), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-eps)), $MachinePrecision] - N[(N[(t$95$1 * t$95$3), $MachinePrecision] * -1.0 + N[(N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + N[(1.0 - (-t$95$3)), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \tan x, -0.5, \frac{\mathsf{fma}\left(-0.3333333333333333, \sin x, \left(0.16666666666666666 \cdot \left(1 - \left(-{\tan x}^{2}\right)\right)\right) \cdot \sin x\right)}{\cos x}\right) \cdot \left(-\varepsilon\right) - \mathsf{fma}\left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right) + 0.16666666666666666\right), \varepsilon, \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \tan x\right), \varepsilon, 1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around 0

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

    1. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + \mathsf{fma}\left(0.3333333333333333, t\_0, t\_0\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}, \tan x + {\tan x}^{3}\right), t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (fma
       eps
       (-
        (+ 0.3333333333333333 (fma 0.3333333333333333 t_0 t_0))
        (* -1.0 (/ (pow (sin x) 4.0) (pow (cos x) 4.0))))
       (+ (tan x) (pow (tan x) 3.0)))
      t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (1.0 + fma(eps, fma(eps, ((0.3333333333333333 + fma(0.3333333333333333, t_0, t_0)) - (-1.0 * (pow(sin(x), 4.0) / pow(cos(x), 4.0)))), (tan(x) + pow(tan(x), 3.0))), t_0));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, Float64(Float64(0.3333333333333333 + fma(0.3333333333333333, t_0, t_0)) - Float64(-1.0 * Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))), Float64(tan(x) + (tan(x) ^ 3.0))), t_0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(eps * N[(N[(0.3333333333333333 + N[(0.3333333333333333 * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(x + \varepsilon\right)} - \tan x \]

   2. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   5. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x + \frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   7. lift-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   8. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   9. lower-tan.f64 N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   11. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   13. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x \]

   14. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   15. lower-tan.f64 62.5

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x \cdot \tan \varepsilon}} - \tan x \]

   3. lift-tan.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \tan \varepsilon} - \tan x \]

   4. lift-tan.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   5. flip3-- N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

   7. metadata-eval N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{\color{blue}{1} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{\color{blue}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   10. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\color{blue}{\tan x} \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   11. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \color{blue}{\tan \varepsilon}\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\color{blue}{\left(\tan x \cdot \tan \varepsilon\right)}}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   13. metadata-eval N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\color{blue}{1} + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{\color{blue}{1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

5. Applied rewrites 62.5%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 + \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \tan x \cdot \tan \varepsilon, 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{1}{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + \mathsf{fma}\left(0.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}, \tan x + {\tan x}^{3}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.3%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.6%

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

Alternative 7: 99.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (tan x) 2.0))))
   (*
    (-
     (fma
      (fma 0.3333333333333333 eps (* 1.0 (/ (* (- 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(fma(0.3333333333333333, eps, (1.0 * (((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(fma(0.3333333333333333, eps, Float64(1.0 * 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[(0.3333333333333333 * eps + N[(1.0 * N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.5%

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

Alternative 8: 99.4% accurate, 0.3× 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 = Float64(-(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 62.3%

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

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 99.4%

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

Alternative 9: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(-{\tan x}^{2}\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, x, 0.6666666666666666 \cdot \varepsilon\right), x, 0.3333333333333333\right) \cdot \varepsilon\right), \varepsilon, t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (- (pow (tan x) 2.0)))))
   (*
    (fma
     (fma
      t_0
      (tan x)
      (*
       (fma
        (fma 1.3333333333333333 x (* 0.6666666666666666 eps))
        x
        0.3333333333333333)
       eps))
     eps
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = 1.0 - -pow(tan(x), 2.0);
	return fma(fma(t_0, tan(x), (fma(fma(1.3333333333333333, x, (0.6666666666666666 * eps)), x, 0.3333333333333333) * eps)), eps, t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(-(tan(x) ^ 2.0)))
	return Float64(fma(fma(t_0, tan(x), Float64(fma(fma(1.3333333333333333, x, Float64(0.6666666666666666 * eps)), x, 0.3333333333333333) * eps)), eps, t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[Tan[x], $MachinePrecision] + N[(N[(N[(1.3333333333333333 * x + N[(0.6666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 99.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), \tan x, \mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, x, 0.6666666666666666 \cdot \varepsilon\right), x, 0.3333333333333333\right) \cdot \varepsilon\right), \varepsilon, 1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]
10. Add Preprocessing

Alternative 10: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{1 \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     (+
      0.3333333333333333
      (* x (fma 0.6666666666666666 eps (* 1.3333333333333333 x))))
     eps
     (* 1.0 (/ (* 1.0 (sin x)) (cos x))))
    eps
    1.0)
   (- (pow (tan x) 2.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma((0.3333333333333333 + (x * fma(0.6666666666666666, eps, (1.3333333333333333 * x)))), eps, (1.0 * ((1.0 * sin(x)) / cos(x)))), eps, 1.0) - -pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(0.3333333333333333 + Float64(x * fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)))), eps, Float64(1.0 * Float64(Float64(1.0 * sin(x)) / cos(x)))), eps, 1.0) - Float64(-(tan(x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.3333333333333333 + N[(x * N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 * N[(N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 99.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{1 \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{1 \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 11: 99.1% accurate, 1.0× 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 62.3%

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

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. tan-quot N/A

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

   9. tan-quot 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 99.1

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

4. Applied rewrites 99.1%

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

Alternative 12: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     (+
      0.3333333333333333
      (* x (fma 0.6666666666666666 eps (* 1.3333333333333333 x))))
     eps
     (* 1.0 (* (fma (* x x) 1.3333333333333333 1.0) x)))
    eps
    1.0)
   (*
    (* x x)
    (-
     (*
      (* x x)
      (-
       (* (* x x) (- (* -0.19682539682539682 (* x x)) 0.37777777777777777))
       0.6666666666666666))
     1.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma((0.3333333333333333 + (x * fma(0.6666666666666666, eps, (1.3333333333333333 * x)))), eps, (1.0 * (fma((x * x), 1.3333333333333333, 1.0) * x))), eps, 1.0) - ((x * x) * (((x * x) * (((x * x) * ((-0.19682539682539682 * (x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(0.3333333333333333 + Float64(x * fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)))), eps, Float64(1.0 * Float64(fma(Float64(x * x), 1.3333333333333333, 1.0) * x))), eps, 1.0) - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.19682539682539682 * Float64(x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.3333333333333333 + N[(x * N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 * N[(N[(N[(x * x), $MachinePrecision] * 1.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.37777777777777777), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 99.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(1 + \frac{4}{3} \cdot {x}^{2}\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(1 + \frac{4}{3} \cdot {x}^{2}\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(\frac{4}{3} \cdot {x}^{2} + 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left({x}^{2} \cdot \frac{4}{3} + 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   8. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

10. Applied rewrites 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    2. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    12. lower--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    14. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot \left(x \cdot x\right) - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    15. lift-*.f64 98.7

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]

13. Applied rewrites 98.7%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 13: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     (+
      0.3333333333333333
      (* x (fma 0.6666666666666666 eps (* 1.3333333333333333 x))))
     eps
     (* 1.0 (* (fma (* x x) 1.3333333333333333 1.0) x)))
    eps
    1.0)
   (*
    (* x x)
    (-
     (* (* x x) (- (* -0.37777777777777777 (* x x)) 0.6666666666666666))
     1.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma((0.3333333333333333 + (x * fma(0.6666666666666666, eps, (1.3333333333333333 * x)))), eps, (1.0 * (fma((x * x), 1.3333333333333333, 1.0) * x))), eps, 1.0) - ((x * x) * (((x * x) * ((-0.37777777777777777 * (x * x)) - 0.6666666666666666)) - 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(0.3333333333333333 + Float64(x * fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)))), eps, Float64(1.0 * Float64(fma(Float64(x * x), 1.3333333333333333, 1.0) * x))), eps, 1.0) - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.37777777777777777 * Float64(x * x)) - 0.6666666666666666)) - 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.3333333333333333 + N[(x * N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 * N[(N[(N[(x * x), $MachinePrecision] * 1.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.37777777777777777 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{2}{3} \cdot \varepsilon + \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 99.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(1 + \frac{4}{3} \cdot {x}^{2}\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(1 + \frac{4}{3} \cdot {x}^{2}\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left(\frac{4}{3} \cdot {x}^{2} + 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\left({x}^{2} \cdot \frac{4}{3} + 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   8. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

10. Applied rewrites 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    2. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + x \cdot \mathsf{fma}\left(\frac{2}{3}, \varepsilon, \frac{4}{3} \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{4}{3}, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot \left(x \cdot x\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 98.7

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]

13. Applied rewrites 98.7%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 + x \cdot \mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), \varepsilon, 1 \cdot \left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 1\right) \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 14: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     0.3333333333333333
     eps
     (* x (+ 1.0 (* 0.6666666666666666 (* eps eps)))))
    eps
    1.0)
   (-
    (*
     (fma
      (fma
       (fma 0.19682539682539682 (* x x) 0.37777777777777777)
       (* x x)
       0.6666666666666666)
      (* x x)
      1.0)
     (* x x))))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + (0.6666666666666666 * (eps * eps))))), eps, 1.0) - -(fma(fma(fma(0.19682539682539682, (x * x), 0.37777777777777777), (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(0.6666666666666666 * Float64(eps * eps))))), eps, 1.0) - Float64(-Float64(fma(fma(fma(0.19682539682539682, Float64(x * x), 0.37777777777777777), Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x)))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(N[(N[(N[(0.19682539682539682 * N[(x * x), $MachinePrecision] + 0.37777777777777777), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.1%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.7%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 15: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     0.3333333333333333
     eps
     (* x (+ 1.0 (* 0.6666666666666666 (* eps eps)))))
    eps
    1.0)
   (-
    (*
     (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0)
     (* x x))))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + (0.6666666666666666 * (eps * eps))))), eps, 1.0) - -(fma(fma(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(0.6666666666666666 * Float64(eps * eps))))), eps, 1.0) - Float64(-Float64(fma(fma(0.37777777777777777, Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x)))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(N[(N[(0.37777777777777777 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.1%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   12. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   13. lower-*.f64 98.6

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.6%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 16: 98.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     0.3333333333333333
     eps
     (* x (+ 1.0 (* 0.6666666666666666 (* eps eps)))))
    eps
    1.0)
   (- (* (fma (* x x) 0.6666666666666666 1.0) (* x x))))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + (0.6666666666666666 * (eps * eps))))), eps, 1.0) - -(fma((x * x), 0.6666666666666666, 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(0.6666666666666666 * Float64(eps * eps))))), eps, 1.0) - Float64(-Float64(fma(Float64(x * x), 0.6666666666666666, 1.0) * Float64(x * x)))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.1%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + \frac{2}{3} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(1 + \frac{2}{3} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left(\frac{2}{3} \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \frac{2}{3} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left({x}^{2}, \frac{2}{3}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   9. lower-*.f64 98.6

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.6%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 17: 98.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-x \cdot x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     0.3333333333333333
     eps
     (* x (+ 1.0 (* 0.6666666666666666 (* eps eps)))))
    eps
    1.0)
   (- (* x x)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + (0.6666666666666666 * (eps * eps))))), eps, 1.0) - -(x * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(0.6666666666666666 * Float64(eps * eps))))), eps, 1.0) - Float64(-Float64(x * x))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(x * x), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.1%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-x \cdot x\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 98.5

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-x \cdot x\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \varepsilon, 1\right) - \left(-x \cdot x\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 18: 98.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* (* eps eps) x)
  (fma (* eps eps) 0.6666666666666666 1.0)
  (* (fma (* eps eps) 0.3333333333333333 1.0) eps)))

C

double code(double x, double eps) {
	return fma(((eps * eps) * x), fma((eps * eps), 0.6666666666666666, 1.0), (fma((eps * eps), 0.3333333333333333, 1.0) * eps));
}

Julia

function code(x, eps)
	return fma(Float64(Float64(eps * eps) * x), fma(Float64(eps * eps), 0.6666666666666666, 1.0), Float64(fma(Float64(eps * eps), 0.3333333333333333, 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.1%

    \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x, \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right)}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right) \cdot \varepsilon\right) \]
11. Add Preprocessing

Alternative 19: 98.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (* (fma (* eps eps) 0.6666666666666666 1.0) x)
    eps
    (* (* eps eps) 0.3333333333333333))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma((fma((eps * eps), 0.6666666666666666, 1.0) * x), eps, ((eps * eps) * 0.3333333333333333)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(fma(Float64(eps * eps), 0.6666666666666666, 1.0) * x), eps, Float64(Float64(eps * eps) * 0.3333333333333333)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * eps + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

10. Applied rewrites 98.1%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 20: 98.1% accurate, 207.0× 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 62.3%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
3. Step-by-step derivation

   1. quot-tan N/A

      \[\leadsto \tan \varepsilon \]

   2. lower-tan.f64 98.1

      \[\leadsto \tan \varepsilon \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\tan \varepsilon} \]

5. Taylor expanded in eps around 0

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

7. Applied rewrites 98.1%

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

Developer Target 1: 99.9% accurate, 0.6× 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: 62.5% 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: 99.1% 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 2025092 
(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 default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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

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

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