2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.5%

Time: 22.0s

Alternatives: 9

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 62.7% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fmm-def 66.0%

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

4. Applied egg-rr 66.0%

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

   1. fmm-undef 66.0%

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

   2. associate-*r/ 66.0%

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

   3. *-rgt-identity 66.0%

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

6. Simplified 66.0%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fmm-undef 99.7%

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

11. Simplified 99.7%

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

    1. expm1-log1p-u 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right)\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

    2. log1p-define 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right)\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

    3. +-commutative 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right)\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}\right)\right) \]

    4. expm1-undefine 99.7%

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

    5. add-exp-log 99.7%

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

    6. unpow2 99.7%

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

    7. unpow2 99.7%

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

    8. frac-times 99.7%

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

    9. tan-quot 99.7%

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

    10. tan-quot 99.7%

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

    11. pow2 99.7%

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

13. Applied egg-rr 99.7%

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

    1. associate--l+ 99.7%

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

    2. metadata-eval 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right)\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \left({\tan x}^{2} + \color{blue}{0}\right)\right) \]

    3. +-rgt-identity 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (-
    1.0
    (*
     eps
     (-
      (*
       eps
       (-
        (- (* (pow (tan x) 2.0) -1.3333333333333333) (pow (tan x) 4.0))
        0.3333333333333333))
      (+ (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 * ((eps * (((pow(tan(x), 2.0) * -1.3333333333333333) - pow(tan(x), 4.0)) - 0.3333333333333333)) - (tan(x) + pow(tan(x), 3.0))))) + (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 - (eps * ((eps * ((((tan(x) ** 2.0d0) * (-1.3333333333333333d0)) - (tan(x) ** 4.0d0)) - 0.3333333333333333d0)) - (tan(x) + (tan(x) ** 3.0d0))))) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

def code(x, eps):
	return eps * ((1.0 - (eps * ((eps * (((math.pow(math.tan(x), 2.0) * -1.3333333333333333) - math.pow(math.tan(x), 4.0)) - 0.3333333333333333)) - (math.tan(x) + math.pow(math.tan(x), 3.0))))) + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((1.0 - (eps * ((eps * ((((tan(x) ^ 2.0) * -1.3333333333333333) - (tan(x) ^ 4.0)) - 0.3333333333333333)) - (tan(x) + (tan(x) ^ 3.0))))) + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 - N[(eps * N[(N[(eps * N[(N[(N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * -1.3333333333333333), $MachinePrecision] - N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $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 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fmm-def 66.0%

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

4. Applied egg-rr 66.0%

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

   1. fmm-undef 66.0%

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

   2. associate-*r/ 66.0%

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

   3. *-rgt-identity 66.0%

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

6. Simplified 66.0%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fmm-undef 99.7%

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

11. Simplified 99.7%

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

13. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right)} - 1\right)} - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
14. Step-by-step derivation

    1. sub-neg 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right)} + \left(-1\right)\right)} - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    2. metadata-eval 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right)} + \color{blue}{-1}\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    3. +-commutative 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right)}\right)} - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    4. log1p-undefine 99.7%

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

    5. rem-exp-log 99.7%

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

    6. associate-+r+ 99.7%

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

    7. metadata-eval 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\left(\color{blue}{0} + \varepsilon \cdot \left(0.3333333333333333 - \mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    8. sub-neg 99.7%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\left(0 + \varepsilon \cdot \color{blue}{\left(0.3333333333333333 + \left(-\mathsf{fma}\left(-1, {\tan x}^{2}, \mathsf{fma}\left(-0.3333333333333333, {\tan x}^{2}, -{\left({\tan x}^{2}\right)}^{2}\right)\right)\right)\right)}\right) - \left(\left(-\tan x\right) - {\tan x}^{3}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + ((eps * ((eps * 0.3333333333333333d0) + (tan(x) + (tan(x) ** 3.0d0)))) + 1.0d0))
end function

Java

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

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + ((eps * ((eps * 0.3333333333333333) + (math.tan(x) + math.pow(math.tan(x), 3.0)))) + 1.0))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + ((eps * ((eps * 0.3333333333333333) + (tan(x) + (tan(x) ^ 3.0)))) + 1.0));
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[(N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fmm-def 66.0%

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

4. Applied egg-rr 66.0%

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

   1. fmm-undef 66.0%

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

   2. associate-*r/ 66.0%

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

   3. *-rgt-identity 66.0%

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

6. Simplified 66.0%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fmm-undef 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in x around 0 99.5%

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

    1. *-commutative 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 1.0));
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[(N[(eps * N[(x + N[(eps * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. tan-sum 66.0%

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

   2. div-inv 66.0%

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

   3. fmm-def 66.0%

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

4. Applied egg-rr 66.0%

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

   1. fmm-undef 66.0%

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

   2. associate-*r/ 66.0%

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

   3. *-rgt-identity 66.0%

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

6. Simplified 66.0%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. fmm-undef 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in x around 0 98.7%

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

    1. unpow2 98.7%

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

    2. associate-*r* 98.7%

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

    3. *-commutative 98.7%

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

    4. distribute-rgt-out 98.7%

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

    5. *-commutative 98.7%

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

14. Simplified 98.7%

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

15. Final simplification 98.7%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(x + \varepsilon \cdot 0.3333333333333333\right) + 1\right)\right) \]
16. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-cube-cbrt 65.6%

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

   2. pow3 65.6%

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

4. Applied egg-rr 65.6%

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

5. Taylor expanded in eps around 0 96.4%

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

   1. sub-neg 96.4%

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

   2. mul-1-neg 96.4%

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

   3. remove-double-neg 96.4%

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

7. Simplified 96.4%

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

   1. rem-cube-cbrt 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. unpow2 98.6%

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

   5. unpow2 98.6%

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

   6. frac-times 98.6%

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

   7. tan-quot 98.6%

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

   8. tan-quot 98.6%

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

   9. pow2 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-cube-cbrt 65.6%

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

   2. pow3 65.6%

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

4. Applied egg-rr 65.6%

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

5. Taylor expanded in eps around 0 98.6%

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

   1. cancel-sign-sub-inv 98.6%

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

   2. metadata-eval 98.6%

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

   3. *-lft-identity 98.6%

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

   4. +-commutative 98.6%

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

7. Simplified 98.6%

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

   1. distribute-rgt-in 98.6%

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

   2. unpow2 98.6%

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

   3. unpow2 98.6%

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

   4. frac-times 98.6%

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

   5. tan-quot 98.6%

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

   6. tan-quot 98.6%

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

   7. pow2 98.6%

      \[\leadsto \color{blue}{{\tan x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon \]

   8. *-un-lft-identity 98.6%

      \[\leadsto {\tan x}^{2} \cdot \varepsilon + \color{blue}{\varepsilon} \]

9. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

10. Final simplification 98.6%

    \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
11. Add Preprocessing

Alternative 7: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.7%

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

   1. add-cube-cbrt 65.6%

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

   2. pow3 65.6%

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

4. Applied egg-rr 65.6%

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

5. Taylor expanded in eps around 0 98.6%

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

   1. cancel-sign-sub-inv 98.6%

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

   2. metadata-eval 98.6%

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

   3. *-lft-identity 98.6%

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

   4. +-commutative 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]

9. Final simplification 97.2%

   \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
10. Add Preprocessing

Alternative 8: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 65.7%

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

3. Taylor expanded in x around 0 96.9%

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

   1. tan-quot 96.9%

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

   2. *-un-lft-identity 96.9%

      \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]

5. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 96.9%

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 9: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

real(8) function code(x, eps)
    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 65.7%

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

3. Taylor expanded in x around 0 96.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

4. Taylor expanded in eps around 0 96.9%

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

5. Final simplification 96.9%

   \[\leadsto \varepsilon \]
6. 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 2024110 
(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)))