2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.8%

Time: 7.4s

Alternatives: 17

Speedup: 76.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\frac{\cos \left(x + \varepsilon\right) + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (+
    1.0
    (*
     (* eps eps)
     (-
      (*
       (* eps eps)
       (- 0.008333333333333333 (* 0.0001984126984126984 (* eps eps))))
      0.16666666666666666))))
  (*
   (- (/ (+ (cos (+ x eps)) (cos (- eps x))) 2.0) (* (sin eps) (sin x)))
   (cos x))))

C

double code(double x, double eps) {
	return (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / ((((cos((x + eps)) + cos((eps - x))) / 2.0) - (sin(eps) * sin(x))) * cos(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 * (1.0d0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333d0 - (0.0001984126984126984d0 * (eps * eps)))) - 0.16666666666666666d0)))) / ((((cos((x + eps)) + cos((eps - x))) / 2.0d0) - (sin(eps) * sin(x))) * cos(x))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / ((((Math.cos((x + eps)) + Math.cos((eps - x))) / 2.0) - (Math.sin(eps) * Math.sin(x))) * Math.cos(x));
}

Python

def code(x, eps):
	return (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / ((((math.cos((x + eps)) + math.cos((eps - x))) / 2.0) - (math.sin(eps) * math.sin(x))) * math.cos(x))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * Float64(Float64(Float64(eps * eps) * Float64(0.008333333333333333 - Float64(0.0001984126984126984 * Float64(eps * eps)))) - 0.16666666666666666)))) / Float64(Float64(Float64(Float64(cos(Float64(x + eps)) + cos(Float64(eps - x))) / 2.0) - Float64(sin(eps) * sin(x))) * cos(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / ((((cos((x + eps)) + cos((eps - x))) / 2.0) - (sin(eps) * sin(x))) * cos(x));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.008333333333333333 - N[(0.0001984126984126984 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(eps - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. cos-sum N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon \cdot \cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \color{blue}{\cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon \cdot \sin x}\right) \cdot \cos x} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon} \cdot \sin x\right) \cdot \cos x} \]

   10. lower-sin.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \color{blue}{\sin x}\right) \cdot \cos x} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon \cdot \cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \color{blue}{\cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   4. cos-mult N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\frac{\cos \left(\varepsilon + x\right) + \cos \left(\varepsilon - x\right)}{2}} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\frac{\cos \left(\varepsilon + x\right) + \cos \left(\varepsilon - x\right)}{2}} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   6. cos-sum-rev N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right) + \cos \left(\varepsilon - x\right)}}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   8. cos-sum-rev N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\color{blue}{\cos \left(\varepsilon + x\right)} + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\color{blue}{\cos \left(\varepsilon + x\right)} + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   10. +-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\cos \color{blue}{\left(x + \varepsilon\right)} + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\cos \color{blue}{\left(x + \varepsilon\right)} + \cos \left(\varepsilon - x\right)}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\frac{\cos \left(x + \varepsilon\right) + \color{blue}{\cos \left(\varepsilon - x\right)}}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   13. lower--.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\frac{\cos \left(x + \varepsilon\right) + \cos \color{blue}{\left(\varepsilon - x\right)}}{2} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

10. Applied rewrites 99.8%

    \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\color{blue}{\frac{\cos \left(x + \varepsilon\right) + \cos \left(\varepsilon - x\right)}{2}} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (+
    1.0
    (*
     (* eps eps)
     (-
      (*
       (* eps eps)
       (- 0.008333333333333333 (* 0.0001984126984126984 (* eps eps))))
      0.16666666666666666))))
  (*
   (-
    (*
     (fma
      (fma
       (fma (* eps eps) -0.001388888888888889 0.041666666666666664)
       (* eps eps)
       -0.5)
      (* eps eps)
      1.0)
     (cos x))
    (* (sin eps) (sin x)))
   (cos x))))

C

double code(double x, double eps) {
	return (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / (((fma(fma(fma((eps * eps), -0.001388888888888889, 0.041666666666666664), (eps * eps), -0.5), (eps * eps), 1.0) * cos(x)) - (sin(eps) * sin(x))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * Float64(Float64(Float64(eps * eps) * Float64(0.008333333333333333 - Float64(0.0001984126984126984 * Float64(eps * eps)))) - 0.16666666666666666)))) / Float64(Float64(Float64(fma(fma(fma(Float64(eps * eps), -0.001388888888888889, 0.041666666666666664), Float64(eps * eps), -0.5), Float64(eps * eps), 1.0) * cos(x)) - Float64(sin(eps) * sin(x))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.008333333333333333 - N[(0.0001984126984126984 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. cos-sum N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon \cdot \cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \color{blue}{\cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon \cdot \sin x}\right) \cdot \cos x} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon} \cdot \sin x\right) \cdot \cos x} \]

   10. lower-sin.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \color{blue}{\sin x}\right) \cdot \cos x} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

9. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    2. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right) \cdot {\varepsilon}^{2} + 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}, \color{blue}{{\varepsilon}^{2}}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    4. negate-sub N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{\varepsilon}}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    5. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \frac{-1}{2}, {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}, {\varepsilon}^{2}, \frac{-1}{2}\right), {\color{blue}{\varepsilon}}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    8. +-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720} \cdot {\varepsilon}^{2} + \frac{1}{24}, {\varepsilon}^{2}, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    9. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{-1}{720} + \frac{1}{24}, {\varepsilon}^{2}, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    10. lower-fma.f64 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{720}, \frac{1}{24}\right), {\varepsilon}^{2}, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    11. pow2 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{720}, \frac{1}{24}\right), {\varepsilon}^{2}, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    12. lift-*.f64 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{720}, \frac{1}{24}\right), {\varepsilon}^{2}, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    13. pow2 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{720}, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{720}, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2}\right), {\varepsilon}^{2}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    15. pow2 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{720}, \frac{1}{24}\right), \varepsilon \cdot \varepsilon, \frac{-1}{2}\right), \varepsilon \cdot \color{blue}{\varepsilon}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

    16. lift-*.f64 99.8

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right), \varepsilon \cdot \color{blue}{\varepsilon}, 1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.001388888888888889, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right), \varepsilon \cdot \varepsilon, 1\right)} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]
12. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (+
    1.0
    (*
     (* eps eps)
     (-
      (*
       (* eps eps)
       (- 0.008333333333333333 (* 0.0001984126984126984 (* eps eps))))
      0.16666666666666666))))
  (*
   (-
    (* (cos eps) (cos x))
    (*
     (*
      (fma
       (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)
       (* eps eps)
       1.0)
      eps)
     (sin x)))
   (cos x))))

C

double code(double x, double eps) {
	return (eps * (1.0 + ((eps * eps) * (((eps * eps) * (0.008333333333333333 - (0.0001984126984126984 * (eps * eps)))) - 0.16666666666666666)))) / (((cos(eps) * cos(x)) - ((fma(fma((eps * eps), 0.008333333333333333, -0.16666666666666666), (eps * eps), 1.0) * eps) * sin(x))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * Float64(Float64(Float64(eps * eps) * Float64(0.008333333333333333 - Float64(0.0001984126984126984 * Float64(eps * eps)))) - 0.16666666666666666)))) / Float64(Float64(Float64(cos(eps) * cos(x)) - Float64(Float64(fma(fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666), Float64(eps * eps), 1.0) * eps) * sin(x))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.008333333333333333 - N[(0.0001984126984126984 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. cos-sum N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon \cdot \cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\color{blue}{\cos \varepsilon} \cdot \cos x - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \color{blue}{\cos x} - \sin \varepsilon \cdot \sin x\right) \cdot \cos x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon \cdot \sin x}\right) \cdot \cos x} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon} \cdot \sin x\right) \cdot \cos x} \]

   10. lower-sin.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \color{blue}{\sin x}\right) \cdot \cos x} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\color{blue}{\left(\cos \varepsilon \cdot \cos x - \sin \varepsilon \cdot \sin x\right)} \cdot \cos x} \]

9. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \sin x\right) \cdot \cos x} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}\right) \cdot \sin x\right) \cdot \cos x} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}\right) \cdot \sin x\right) \cdot \cos x} \]

    3. +-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    4. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\left(\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot {\varepsilon}^{2} + 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    6. negate-sub N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    7. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    9. lower-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    10. pow2 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    11. lift-*.f64 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    12. pow2 N/A

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

    13. lift-*.f64 99.8

        \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \sin x\right) \cdot \cos x} \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\left(\cos \varepsilon \cdot \cos x - \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right)} \cdot \sin x\right) \cdot \cos x} \]
12. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right), \varepsilon \cdot \varepsilon, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (fma
     (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
     (* eps eps)
     -0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (* (cos (+ x eps)) (cos x))))

C

double code(double x, double eps) {
	return (fma(fma(fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333), (eps * eps), -0.16666666666666666), (eps * eps), 1.0) * eps) / (cos((x + eps)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333), Float64(eps * eps), -0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(x + eps)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right), \varepsilon \cdot \varepsilon, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cos x}} \]
9. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (* (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	return (fma(fma((eps * eps), 0.008333333333333333, -0.16666666666666666), (eps * eps), 1.0) * eps) / (cos((eps + x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) \cdot {\varepsilon}^{2} + 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. negate-sub N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\varepsilon}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), {\varepsilon}^{2}, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lift-*.f64 99.7

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
10. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* eps (- 1.0 (* 0.16666666666666666 (* eps eps))))
  (* (sin (+ (+ x eps) (/ PI 2.0))) (cos x))))

C

double code(double x, double eps) {
	return (eps * (1.0 - (0.16666666666666666 * (eps * eps)))) / (sin(((x + eps) + (((double) M_PI) / 2.0))) * cos(x));
}

Java

public static double code(double x, double eps) {
	return (eps * (1.0 - (0.16666666666666666 * (eps * eps)))) / (Math.sin(((x + eps) + (Math.PI / 2.0))) * Math.cos(x));
}

Python

def code(x, eps):
	return (eps * (1.0 - (0.16666666666666666 * (eps * eps)))) / (math.sin(((x + eps) + (math.pi / 2.0))) * math.cos(x))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 - Float64(0.16666666666666666 * Float64(eps * eps)))) / Float64(sin(Float64(Float64(x + eps) + Float64(pi / 2.0))) * cos(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 - (0.16666666666666666 * (eps * eps)))) / (sin(((x + eps) + (pi / 2.0))) * cos(x));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 - N[(0.16666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(N[(x + eps), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\sin \left(\left(x + \varepsilon\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos x} \]

8. Applied rewrites 99.7%

   \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\color{blue}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right)} \cdot \cos x} \]
9. Add Preprocessing

Alternative 7: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right)}}{\cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/ (* (fma -0.16666666666666666 (* eps eps) 1.0) eps) (cos (+ x eps)))
  (cos x)))

C

double code(double x, double eps) {
	return ((fma(-0.16666666666666666, (eps * eps), 1.0) * eps) / cos((x + eps))) / cos(x);
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / cos(Float64(x + eps))) / cos(x))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right)}}{\cos x}} \]
9. Add Preprocessing

Alternative 8: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma -0.16666666666666666 (* eps eps) 1.0) eps)
  (* (cos (+ x eps)) (cos x))))

C

double code(double x, double eps) {
	return (fma(-0.16666666666666666, (eps * eps), 1.0) * eps) / (cos((x + eps)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(x + eps)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 99.7

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

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \mathsf{Rewrite=>}\left(lift-+.f64, \left(\varepsilon + x\right)\right) \cdot \cos x} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \mathsf{Rewrite=>}\left(+-commutative, \left(x + \varepsilon\right)\right) \cdot \cos x} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \mathsf{Rewrite=>}\left(lower-+.f64, \left(x + \varepsilon\right)\right) \cdot \cos x} \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cos x}} \]
9. Add Preprocessing

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (eps / cos((x + eps))) / cos(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 / cos((x + eps))) / cos(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in eps around 0

   \[\leadsto \frac{\frac{\varepsilon}{\cos \left(x + \varepsilon\right)}}{\cos x} \]
10. Step-by-step derivation

11. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{\varepsilon}{\cos \left(x + \varepsilon\right)}}{\cos x} \]
12. Add Preprocessing

Alternative 10: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (cos((eps + x)) * cos(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 / (cos((eps + x)) * cos(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
7. Add Preprocessing

Alternative 11: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* eps (- 1.0 (* 0.16666666666666666 (* eps eps))))
  (fma (cos (+ x x)) 0.5 0.5)))

C

double code(double x, double eps) {
	return (eps * (1.0 - (0.16666666666666666 * (eps * eps)))) / fma(cos((x + x)), 0.5, 0.5);
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 - Float64(0.16666666666666666 * Float64(eps * eps)))) / fma(cos(Float64(x + x)), 0.5, 0.5))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 - N[(0.16666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

      \[\leadsto \frac{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 - 0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon \cdot \left(1 - \frac{1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{\color{blue}{{\cos x}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 N/A

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

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f64 98.9

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

9. Applied rewrites 98.9%

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

Alternative 12: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (fma (cos (+ x x)) 0.5 0.5)))

C

double code(double x, double eps) {
	return eps / fma(cos((x + x)), 0.5, 0.5);
}

Julia

function code(x, eps)
	return Float64(eps / fma(cos(Float64(x + x)), 0.5, 0.5))
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 62.4%

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

4. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right)} - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{6}}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \left(\mathsf{neg}\left(\frac{-1}{5040}\right)\right) \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} - \frac{1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{6}\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   15. lower-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 - 0.0001984126984126984 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.16666666666666666\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\varepsilon}{\color{blue}{{\cos x}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\varepsilon}{\cos x \cdot \color{blue}{\cos x}} \]

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 98.9

      \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} \]

9. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)}} \]
10. Add Preprocessing

Alternative 13: 98.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

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%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 98.3

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

10. Applied rewrites 98.3%

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

Alternative 14: 98.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

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%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 98.3

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

10. Applied rewrites 98.3%

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

Alternative 15: 98.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

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%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 16: 98.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

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%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 98.1

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

10. Applied rewrites 98.1%

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

Alternative 17: 97.7% accurate, 76.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.4%

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

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 97.7

      \[\leadsto \tan \varepsilon \]

4. Applied rewrites 97.7%

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

5. Taylor expanded in eps around 0

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

7. Applied rewrites 97.7%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% 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 2025110 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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

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

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