2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.6%

Time: 18.2s

Alternatives: 11

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

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 = sin(x) ** 2.0d0
    t_1 = t_0 * (cos(x) ** (-2.0d0))
    code = eps * (((eps * ((eps * (0.3333333333333333d0 + (t_1 - (((-0.3333333333333333d0) * t_1) - ((sin(x) ** 4.0d0) * (cos(x) ** (-4.0d0))))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))) + 1.0d0) + (t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

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

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.6%

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

    1. +-commutative 99.6%

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

    2. *-un-lft-identity 99.6%

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

    3. fma-define 99.6%

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

11. Applied egg-rr 99.6%

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

    1. fma-undefine 99.6%

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

    2. unsub-neg 99.6%

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

    3. *-lft-identity 99.6%

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

    4. fma-undefine 99.6%

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

    5. unsub-neg 99.6%

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

    6. *-commutative 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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 = sin(x) ** 2.0d0
    t_1 = t_0 / (cos(x) ** 2.0d0)
    code = eps * (t_1 + ((eps * ((eps * (0.3333333333333333d0 + (t_1 + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * t_0))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Taylor expanded in x around 0 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * ((t_0 + (eps * ((eps * (0.3333333333333333 + (t_0 + ((sin(x) ^ 4.0) / (cos(x) ^ 4.0))))) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))) + 1.0);
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[(t$95$0 + N[(eps * N[(N[(eps * N[(0.3333333333333333 + N[(t$95$0 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 61.3%

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

6. Taylor expanded in eps around 0 99.6%

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

   1. associate--l+ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))) + (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)) + 1.0d0) / 2.0d0)) + ((eps * (((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))) + (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.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))) + (eps * 0.3333333333333333))) + 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)) + ((eps * (((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))) + (eps * 0.3333333333333333))) + 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)) + Float64(Float64(eps * Float64(Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))) + Float64(eps * 0.3333333333333333))) + 1.0)))
end

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

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

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(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)) + 1.0d0) / 2.0d0)) + ((eps * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(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.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))) + 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)) + ((eps * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)))) + 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)) + Float64(Float64(eps * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))) + 1.0)))
end

MATLAB

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

Derivation

1. Initial program 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

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

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in eps around 0 99.5%

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

    1. mul-1-neg 99.5%

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

    2. distribute-rgt-neg-in 99.5%

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

    3. distribute-lft-out 99.5%

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

    4. distribute-lft-neg-in 99.5%

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

    5. metadata-eval 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 6: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / ((cos((x * 2.0)) + 1.0) / 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)) + 1.0d0) / 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.cos((x * 2.0)) + 1.0) / 2.0)) + ((eps * (x + (eps * 0.3333333333333333))) + 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)) + ((eps * (x + (eps * 0.3333333333333333))) + 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)) + 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)) + 1.0) / 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[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $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 61.4%

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

   1. tan-sum 61.6%

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

   2. div-inv 61.6%

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

   3. fma-neg 61.5%

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

4. Applied egg-rr 61.5%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

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

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in x around 0 99.3%

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

    1. +-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 7: 99.0% 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 61.4%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. mul-1-neg 99.1%

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

   3. remove-double-neg 99.1%

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

5. Simplified 99.1%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

7. Applied egg-rr 99.1%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

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

   3. cos-0 99.6%

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

   4. count-2 99.6%

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

   5. *-commutative 99.6%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 98.8%

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

5. Taylor expanded in eps around 0 98.8%

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

   1. unpow2 98.8%

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

   2. distribute-rgt-out 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 9: 98.4% 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 61.4%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. mul-1-neg 99.1%

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

   3. remove-double-neg 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.6%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
7. Add Preprocessing

Alternative 10: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. mul-1-neg 99.1%

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

   3. remove-double-neg 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.6%

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

   1. *-rgt-identity 98.6%

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

   2. distribute-lft-out 98.6%

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

   3. +-commutative 98.6%

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

   4. unpow2 98.6%

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

   5. fma-define 98.6%

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

8. Simplified 98.6%

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

Alternative 11: 98.0% 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 61.4%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. mul-1-neg 99.1%

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

   3. remove-double-neg 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.2%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024117 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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