2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.5%

Time: 23.1s

Alternatives: 9

Speedup: 1.9×

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

real(8) function code(x, eps)
    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: 62.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + {\tan x}^{4}\right) - {\tan x}^{2} \cdot -1.3333333333333333, \tan x + {\tan x}^{3}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    1.0
    (*
     eps
     (fma
      eps
      (-
       (+ 0.3333333333333333 (pow (tan x) 4.0))
       (* (pow (tan x) 2.0) -1.3333333333333333))
      (+ (tan x) (pow (tan x) 3.0)))))
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * ((1.0 + (eps * fma(eps, ((0.3333333333333333 + pow(tan(x), 4.0)) - (pow(tan(x), 2.0) * -1.3333333333333333)), (tan(x) + pow(tan(x), 3.0))))) + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * fma(eps, Float64(Float64(0.3333333333333333 + (tan(x) ^ 4.0)) - Float64((tan(x) ^ 2.0) * -1.3333333333333333)), Float64(tan(x) + (tan(x) ^ 3.0))))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(eps * N[(eps * N[(N[(0.3333333333333333 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

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

   1. tan-sum 61.8%

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

   2. clear-num 61.4%

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

4. Applied egg-rr 61.4%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - -1 \cdot \left(\tan x + {\tan x}^{3}\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Step-by-step derivation

   1. unpow1 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. distribute-lft-neg-in 99.6%

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

9. Simplified 99.6%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, {\sin x}^{2} \cdot \left({\cos x}^{-2} \cdot -0.3333333333333333\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right), \tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
10. Step-by-step derivation

    1. sub-neg 99.6%

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

11. Applied egg-rr 99.6%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 + \left(-\mathsf{fma}\left(-1, {\tan x}^{2}, -0.3333333333333333 \cdot {\tan x}^{2} - {\tan x}^{4}\right)\right)}, \tan x + {\tan x}^{3}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
12. Step-by-step derivation

    1. sub-neg 99.6%

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

    2. fma-undefine 99.6%

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

    3. neg-mul-1 99.6%

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

    4. associate-+r- 99.6%

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

    5. sub-neg 99.6%

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

    6. +-commutative 99.6%

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

    7. associate--r+ 99.6%

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

    8. metadata-eval 99.6%

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

    9. pow-sqr 99.6%

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

    10. distribute-lft-neg-out 99.6%

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

    11. cancel-sign-sub 99.6%

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

    12. pow-sqr 99.6%

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

    13. metadata-eval 99.6%

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

13. Simplified 99.6%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(0.3333333333333333 + {\tan x}^{4}\right) - {\tan x}^{2} \cdot -1.3333333333333333}, \tan x + {\tan x}^{3}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

14. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \left(0.3333333333333333 + {\tan x}^{4}\right) - {\tan x}^{2} \cdot -1.3333333333333333, \tan x + {\tan x}^{3}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
15. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(eps * N[(eps * 0.3333333333333333 + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

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

   1. tan-sum 61.8%

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

   2. clear-num 61.4%

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

4. Applied egg-rr 61.4%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right)\right) - -1 \cdot \left(\tan x + {\tan x}^{3}\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Step-by-step derivation

   1. unpow1 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. distribute-lft-neg-in 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left({\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + {\varepsilon}^{2} \cdot 1.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* eps (+ 1.0 (* 0.3333333333333333 (pow eps 2.0))))
  (*
   x
   (+
    (pow eps 2.0)
    (* eps (* x (+ 1.0 (* (pow eps 2.0) 1.3333333333333333))))))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * pow(eps, 2.0)))) + (x * (pow(eps, 2.0) + (eps * (x * (1.0 + (pow(eps, 2.0) * 1.3333333333333333))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (1.0d0 + (0.3333333333333333d0 * (eps ** 2.0d0)))) + (x * ((eps ** 2.0d0) + (eps * (x * (1.0d0 + ((eps ** 2.0d0) * 1.3333333333333333d0))))))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * Math.pow(eps, 2.0)))) + (x * (Math.pow(eps, 2.0) + (eps * (x * (1.0 + (Math.pow(eps, 2.0) * 1.3333333333333333))))));
}

Python

def code(x, eps):
	return (eps * (1.0 + (0.3333333333333333 * math.pow(eps, 2.0)))) + (x * (math.pow(eps, 2.0) + (eps * (x * (1.0 + (math.pow(eps, 2.0) * 1.3333333333333333))))))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * (eps ^ 2.0)))) + Float64(x * Float64((eps ^ 2.0) + Float64(eps * Float64(x * Float64(1.0 + Float64((eps ^ 2.0) * 1.3333333333333333)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + (0.3333333333333333 * (eps ^ 2.0)))) + (x * ((eps ^ 2.0) + (eps * (x * (1.0 + ((eps ^ 2.0) * 1.3333333333333333))))));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Power[eps, 2.0], $MachinePrecision] + N[(eps * N[(x * N[(1.0 + N[(N[Power[eps, 2.0], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2}\right)} \]

5. Final simplification 97.8%

   \[\leadsto \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left({\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + {\varepsilon}^{2} \cdot 1.3333333333333333\right)\right)\right) \]
6. Add Preprocessing

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + {\varepsilon}^{2} \cdot 1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (+
    (* 0.3333333333333333 (pow eps 2.0))
    (* x (+ eps (* x (+ 1.0 (* (pow eps 2.0) 1.3333333333333333)))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((0.3333333333333333 * pow(eps, 2.0)) + (x * (eps + (x * (1.0 + (pow(eps, 2.0) * 1.3333333333333333)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((0.3333333333333333d0 * (eps ** 2.0d0)) + (x * (eps + (x * (1.0d0 + ((eps ** 2.0d0) * 1.3333333333333333d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((0.3333333333333333 * Math.pow(eps, 2.0)) + (x * (eps + (x * (1.0 + (Math.pow(eps, 2.0) * 1.3333333333333333)))))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((0.3333333333333333 * math.pow(eps, 2.0)) + (x * (eps + (x * (1.0 + (math.pow(eps, 2.0) * 1.3333333333333333)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(x * Float64(eps + Float64(x * Float64(1.0 + Float64((eps ^ 2.0) * 1.3333333333333333))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((0.3333333333333333 * (eps ^ 2.0)) + (x * (eps + (x * (1.0 + ((eps ^ 2.0) * 1.3333333333333333)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps + N[(x * N[(1.0 + N[(N[Power[eps, 2.0], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]

5. Final simplification 97.8%

   \[\leadsto \varepsilon \cdot \left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + {\varepsilon}^{2} \cdot 1.3333333333333333\right)\right)\right)\right) \]
6. Add Preprocessing

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.4%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Add Preprocessing

Alternative 7: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* 0.3333333333333333 (pow eps 3.0))))

C

double code(double x, double eps) {
	return eps + (0.3333333333333333 * pow(eps, 3.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (0.3333333333333333d0 * (eps ** 3.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (0.3333333333333333 * Math.pow(eps, 3.0));
}

Python

def code(x, eps):
	return eps + (0.3333333333333333 * math.pow(eps, 3.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(0.3333333333333333 * (eps ^ 3.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (0.3333333333333333 * (eps ^ 3.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(0.3333333333333333 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-in 97.4%

      \[\leadsto \color{blue}{1 \cdot \varepsilon + \left(0.3333333333333333 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon} \]

   2. *-lft-identity 97.4%

      \[\leadsto \color{blue}{\varepsilon} + \left(0.3333333333333333 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   3. associate-*l* 97.4%

      \[\leadsto \varepsilon + \color{blue}{0.3333333333333333 \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)} \]

   4. unpow2 97.4%

      \[\leadsto \varepsilon + 0.3333333333333333 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \varepsilon\right) \]

   5. unpow3 97.4%

      \[\leadsto \varepsilon + 0.3333333333333333 \cdot \color{blue}{{\varepsilon}^{3}} \]

6. Simplified 97.4%

   \[\leadsto \color{blue}{\varepsilon + 0.3333333333333333 \cdot {\varepsilon}^{3}} \]
7. Add Preprocessing

Alternative 8: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 59.9%

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

4. Final simplification 59.9%

   \[\leadsto \tan \left(\varepsilon + x\right) - x \]
5. Add Preprocessing

Alternative 9: 7.9% accurate, 102.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := (-x)

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 59.9%

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

4. Taylor expanded in x around inf 8.0%

   \[\leadsto \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. neg-mul-1 8.0%

      \[\leadsto \color{blue}{-x} \]

6. Simplified 8.0%

   \[\leadsto \color{blue}{-x} \]
7. Add Preprocessing

Developer target: 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

real(8) function code(x, eps)
    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]

Reproduce

herbie shell --seed 2024111 
(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
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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