2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.6%

Time: 33.0s

Alternatives: 10

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: 62.2% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{{\sin x}^{2}}{t\_0}\\ \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(\left(t\_1 \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) - t\_1\right) - 0.3333333333333333\right)\right) + 1\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (/ (pow (sin x) 2.0) t_0)))
   (*
    eps
    (+
     (+
      (*
       eps
       (-
        (+ (tan x) (pow (tan x) 3.0))
        (*
         eps
         (-
          (-
           (-
            (* t_1 -0.3333333333333333)
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
           t_1)
          0.3333333333333333))))
      1.0)
     (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0) / t_0;
	return eps * (((eps * ((tan(x) + pow(tan(x), 3.0)) - (eps * ((((t_1 * -0.3333333333333333) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))) - t_1) - 0.3333333333333333)))) + 1.0) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = cos(x) ** 2.0d0
    t_1 = (sin(x) ** 2.0d0) / t_0
    code = eps * (((eps * ((tan(x) + (tan(x) ** 3.0d0)) - (eps * ((((t_1 * (-0.3333333333333333d0)) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0))) - t_1) - 0.3333333333333333d0)))) + 1.0d0) + ((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.cos(x), 2.0);
	double t_1 = Math.pow(Math.sin(x), 2.0) / t_0;
	return eps * (((eps * ((Math.tan(x) + Math.pow(Math.tan(x), 3.0)) - (eps * ((((t_1 * -0.3333333333333333) - (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0))) - t_1) - 0.3333333333333333)))) + 1.0) + ((0.5 - (Math.cos((x * 2.0)) / 2.0)) / t_0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	t_1 = math.pow(math.sin(x), 2.0) / t_0
	return eps * (((eps * ((math.tan(x) + math.pow(math.tan(x), 3.0)) - (eps * ((((t_1 * -0.3333333333333333) - (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0))) - t_1) - 0.3333333333333333)))) + 1.0) + ((0.5 - (math.cos((x * 2.0)) / 2.0)) / t_0))

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64((sin(x) ^ 2.0) / t_0)
	return Float64(eps * Float64(Float64(Float64(eps * Float64(Float64(tan(x) + (tan(x) ^ 3.0)) - Float64(eps * Float64(Float64(Float64(Float64(t_1 * -0.3333333333333333) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1) - 0.3333333333333333)))) + 1.0) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	t_1 = (sin(x) ^ 2.0) / t_0;
	tmp = eps * (((eps * ((tan(x) + (tan(x) ^ 3.0)) - (eps * ((((t_1 * -0.3333333333333333) - ((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1) - 0.3333333333333333)))) + 1.0) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, N[(eps * N[(N[(N[(eps * N[(N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 61.0%

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

   1. tan-sum 61.0%

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

   2. div-inv 61.0%

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

   3. fmm-def 61.0%

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

4. Applied egg-rr 61.0%

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

   1. fmm-undef 61.0%

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

   2. *-commutative 61.0%

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

   3. associate-/r/ 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in eps around 0 100.0%

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

   1. tan-quot 100.0%

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

   2. distribute-lft-out 100.0%

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

   3. add-sqr-sqrt 46.9%

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

   4. sqrt-unprod 99.6%

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

   5. *-un-lft-identity 99.6%

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

   6. metadata-eval 99.6%

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

   7. swap-sqr 99.6%

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

   8. sqrt-unprod 52.8%

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

   9. add-sqr-sqrt 30.9%

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

   10. sqrt-unprod 52.7%

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

   11. sqr-neg 52.7%

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

   12. mul-1-neg 52.7%

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

   13. mul-1-neg 52.7%

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

9. Applied egg-rr 100.0%

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

    1. neg-mul-1 100.0%

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

    2. distribute-neg-in 100.0%

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

    3. unsub-neg 100.0%

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

11. Simplified 100.0%

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

    1. unpow2 100.0%

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

    2. sin-mult 100.0%

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

13. Applied egg-rr 100.0%

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

    1. div-sub 100.0%

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

    2. +-inverses 100.0%

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

    3. cos-0 100.0%

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

    4. metadata-eval 100.0%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

    5. count-2 100.0%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right) \]

    6. *-commutative 100.0%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right) \]

15. Simplified 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right) \]

16. Final simplification 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - 0.3333333333333333\right)\right) + 1\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]
17. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

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)) - ((-1.0d0) - (eps * ((eps * 0.3333333333333333d0) + (tan(x) + (tan(x) ** 3.0d0))))))
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)) - (-1.0 - (eps * ((eps * 0.3333333333333333) + (Math.tan(x) + Math.pow(Math.tan(x), 3.0))))));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (-1.0 - (eps * ((eps * 0.3333333333333333) + (math.tan(x) + math.pow(math.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 * Float64(Float64(eps * 0.3333333333333333) + Float64(tan(x) + (tan(x) ^ 3.0)))))))
end

MATLAB

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

Derivation

1. Initial program 61.0%

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

   1. tan-sum 61.0%

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

   2. div-inv 61.0%

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

   3. fmm-def 61.0%

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

4. Applied egg-rr 61.0%

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

   1. fmm-undef 61.0%

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

   2. *-commutative 61.0%

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

   3. associate-/r/ 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in eps around 0 100.0%

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

   1. tan-quot 100.0%

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

   2. distribute-lft-out 100.0%

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

   3. add-sqr-sqrt 46.9%

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

   4. sqrt-unprod 99.6%

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

   5. *-un-lft-identity 99.6%

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

   6. metadata-eval 99.6%

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

   7. swap-sqr 99.6%

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

   8. sqrt-unprod 52.8%

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

   9. add-sqr-sqrt 30.9%

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

   10. sqrt-unprod 52.7%

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

   11. sqr-neg 52.7%

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

   12. mul-1-neg 52.7%

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

   13. mul-1-neg 52.7%

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

9. Applied egg-rr 100.0%

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

    1. neg-mul-1 100.0%

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

    2. distribute-neg-in 100.0%

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

    3. unsub-neg 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around 0 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) (pow (cos x) 2.0))
   (+ (+ (* 0.3333333333333333 (pow eps 2.0)) (* eps x)) 1.0))))

C

double code(double x, double eps) {
	return eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / pow(cos(x), 2.0)) + (((0.3333333333333333 * pow(eps, 2.0)) + (eps * x)) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / (cos(x) ** 2.0d0)) + (((0.3333333333333333d0 * (eps ** 2.0d0)) + (eps * x)) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (((0.5 - (Math.cos((x * 2.0)) / 2.0)) / Math.pow(Math.cos(x), 2.0)) + (((0.3333333333333333 * Math.pow(eps, 2.0)) + (eps * x)) + 1.0));
}

Python

def code(x, eps):
	return eps * (((0.5 - (math.cos((x * 2.0)) / 2.0)) / math.pow(math.cos(x), 2.0)) + (((0.3333333333333333 * math.pow(eps, 2.0)) + (eps * x)) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / (cos(x) ^ 2.0)) + Float64(Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(eps * x)) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / (cos(x) ^ 2.0)) + (((0.3333333333333333 * (eps ^ 2.0)) + (eps * x)) + 1.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Taylor expanded in x around 0 99.8%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

6. Applied egg-rr 99.8%

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

   1. div-sub 100.0%

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

   2. +-inverses 100.0%

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

   3. cos-0 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}\right) \]

   5. count-2 100.0%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}\right) \]

   6. *-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}\right) \]

8. Simplified 99.8%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)\right) - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right) \]

9. Final simplification 99.8%

   \[\leadsto \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \left(\left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right) + 1\right)\right) \]
10. Add Preprocessing

Alternative 4: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)) + 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)) + 1.0d0) / 2.0d0)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. remove-double-neg 99.6%

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

5. Simplified 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Taylor expanded in x around 0 99.8%

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

5. Taylor expanded in x around 0 99.4%

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

6. Taylor expanded in x around 0 99.4%

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

7. Final simplification 99.4%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* (pow x 2.0) (+ eps (* (pow x 2.0) (* eps 0.6666666666666666))))))

C

double code(double x, double eps) {
	return eps + (pow(x, 2.0) * (eps + (pow(x, 2.0) * (eps * 0.6666666666666666))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((x ** 2.0d0) * (eps + ((x ** 2.0d0) * (eps * 0.6666666666666666d0))))
end function

Java

public static double code(double x, double eps) {
	return eps + (Math.pow(x, 2.0) * (eps + (Math.pow(x, 2.0) * (eps * 0.6666666666666666))));
}

Python

def code(x, eps):
	return eps + (math.pow(x, 2.0) * (eps + (math.pow(x, 2.0) * (eps * 0.6666666666666666))))

Julia

function code(x, eps)
	return Float64(eps + Float64((x ^ 2.0) * Float64(eps + Float64((x ^ 2.0) * Float64(eps * 0.6666666666666666)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((x ^ 2.0) * (eps + ((x ^ 2.0) * (eps * 0.6666666666666666))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. remove-double-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 99.4%

      \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + \color{blue}{\left(0.6666666666666666 \cdot \varepsilon\right) \cdot {x}^{2}}\right) \]

   2. *-commutative 99.4%

      \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + \color{blue}{\left(\varepsilon \cdot 0.6666666666666666\right)} \cdot {x}^{2}\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + \left(\varepsilon \cdot 0.6666666666666666\right) \cdot {x}^{2}\right)} \]

9. Final simplification 99.4%

   \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / (1.0 - (x * x))) + 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) / (1.0d0 - (x * x))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. remove-double-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. unsub-neg 99.4%

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

8. Simplified 99.4%

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

   1. unpow2 99.3%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (pow(sin(x), 2.0) + ((eps * x) + 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) + ((eps * x) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Taylor expanded in x around 0 99.8%

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

5. Taylor expanded in x around 0 99.4%

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

6. Taylor expanded in eps around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. neg-mul-1 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 9: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. remove-double-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 99.3%

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

8. Simplified 99.3%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \varepsilon} \]
9. Step-by-step derivation

   1. unpow2 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 10: 97.8% 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 61.0%

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

3. Taylor expanded in eps around 0 99.6%

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

   1. sub-neg 99.6%

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

   2. mul-1-neg 99.6%

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

   3. remove-double-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 98.7%

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

7. Final simplification 98.7%

   \[\leadsto \varepsilon \]
8. 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 2024095 
(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)))