2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.6%

Time: 28.0s

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

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

FPCore

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

C

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

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 = (0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_0
    code = eps * (t_1 - ((-1.0d0) - (eps * ((eps * (0.3333333333333333d0 + (((sin(x) ** 2.0d0) / t_0) + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * t_1))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, N[(eps * N[(t$95$1 - N[(-1.0 - N[(eps * N[(N[(eps * N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] + 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$1), $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]), $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. fma-neg 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. fma-neg 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-*l/ 61.0%

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

   4. *-lft-identity 61.0%

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

6. Simplified 61.0%

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

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. unpow2 100.0%

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

   2. sin-mult 100.0%

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

9. Applied egg-rr 100.0%

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

    1. div-sub 100.0%

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

    2. +-inverses 100.0%

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

    3. cos-0 100.0%

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

    4. metadata-eval 100.0%

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

    5. count-2 100.0%

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

    6. *-commutative 100.0%

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

11. Simplified 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}\right) \]
12. Step-by-step derivation

    1. unpow2 100.0%

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

    2. sin-mult 100.0%

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

13. Applied egg-rr 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]
14. Step-by-step derivation

    1. div-sub 100.0%

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

    2. +-inverses 100.0%

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

    3. cos-0 100.0%

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

    4. metadata-eval 100.0%

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

    5. count-2 100.0%

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

    6. *-commutative 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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)
    code = (eps * (t_0 - (((eps ** 2.0d0) * (((-0.3333333333333333d0) * t_0) - (0.3333333333333333d0 * cos(x)))) - cos(x)))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 - N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * t$95$0), $MachinePrecision] - N[(0.3333333333333333 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. metadata-eval 99.9%

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

   3. *-lft-identity 99.9%

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

7. Simplified 99.9%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

9. Applied egg-rr 99.9%

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

    1. div-sub 100.0%

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

    2. +-inverses 100.0%

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

    3. cos-0 100.0%

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

    4. metadata-eval 100.0%

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

    5. count-2 100.0%

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

    6. *-commutative 100.0%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 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. 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. fma-neg 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. fma-neg 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-*l/ 61.0%

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

   4. *-lft-identity 61.0%

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

6. Simplified 61.0%

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

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. unpow2 100.0%

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

   2. sin-mult 100.0%

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

9. Applied egg-rr 100.0%

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

    1. div-sub 100.0%

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

    2. +-inverses 100.0%

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

    3. cos-0 100.0%

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

    4. metadata-eval 100.0%

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

    5. count-2 100.0%

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

    6. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in x around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 5: 98.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (* eps (- (cos x) (/ (- (* 0.5 (cos (* x 2.0))) 0.5) (cos x)))) (cos x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[Cos[x], $MachinePrecision] - N[(N[(N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.6%

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

   1. unpow2 100.0%

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

   2. sin-mult 100.0%

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

7. Applied egg-rr 99.6%

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

   1. div-sub 100.0%

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

   2. +-inverses 100.0%

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

   3. cos-0 100.0%

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

   4. metadata-eval 100.0%

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

   5. count-2 100.0%

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

   6. *-commutative 100.0%

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

9. Simplified 99.6%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x - -1 \cdot \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{\cos x}\right)}{\cos x} \]
10. Step-by-step derivation

    1. *-commutative 99.6%

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

    2. cancel-sign-sub-inv 99.6%

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

    3. metadata-eval 99.6%

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

    4. *-un-lft-identity 99.6%

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

    5. div-inv 99.6%

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

    6. metadata-eval 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x - \frac{0.5 \cdot \cos \left(x \cdot 2\right) - 0.5}{\cos x}\right)}{\cos x} \]
13. Add Preprocessing

Alternative 6: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 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. Final simplification 99.6%

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

Alternative 7: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.5\right)\\ \varepsilon + {x}^{2} \cdot \left(\left(\varepsilon \cdot 0.5 + {x}^{2} \cdot \left(\left(\varepsilon \cdot 0.20833333333333334 - {x}^{2} \cdot \left(\left(-0.5 \cdot \left(\varepsilon \cdot 0.20833333333333334 - \left(\varepsilon \cdot 0.041666666666666664 - t\_0\right)\right) + \left(\varepsilon \cdot -0.001388888888888889 + \left(\varepsilon \cdot 0.5 - \varepsilon \cdot -0.5\right) \cdot 0.041666666666666664\right)\right) - \varepsilon \cdot 0.08472222222222223\right)\right) + \left(t\_0 - \varepsilon \cdot 0.041666666666666664\right)\right)\right) - \varepsilon \cdot -0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* -0.5 (- (* eps -0.5) (* eps 0.5)))))
   (+
    eps
    (*
     (pow x 2.0)
     (-
      (+
       (* eps 0.5)
       (*
        (pow x 2.0)
        (+
         (-
          (* eps 0.20833333333333334)
          (*
           (pow x 2.0)
           (-
            (+
             (*
              -0.5
              (-
               (* eps 0.20833333333333334)
               (- (* eps 0.041666666666666664) t_0)))
             (+
              (* eps -0.001388888888888889)
              (* (- (* eps 0.5) (* eps -0.5)) 0.041666666666666664)))
            (* eps 0.08472222222222223))))
         (- t_0 (* eps 0.041666666666666664)))))
      (* eps -0.5))))))

C

double code(double x, double eps) {
	double t_0 = -0.5 * ((eps * -0.5) - (eps * 0.5));
	return eps + (pow(x, 2.0) * (((eps * 0.5) + (pow(x, 2.0) * (((eps * 0.20833333333333334) - (pow(x, 2.0) * (((-0.5 * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - t_0))) + ((eps * -0.001388888888888889) + (((eps * 0.5) - (eps * -0.5)) * 0.041666666666666664))) - (eps * 0.08472222222222223)))) + (t_0 - (eps * 0.041666666666666664))))) - (eps * -0.5)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (-0.5d0) * ((eps * (-0.5d0)) - (eps * 0.5d0))
    code = eps + ((x ** 2.0d0) * (((eps * 0.5d0) + ((x ** 2.0d0) * (((eps * 0.20833333333333334d0) - ((x ** 2.0d0) * ((((-0.5d0) * ((eps * 0.20833333333333334d0) - ((eps * 0.041666666666666664d0) - t_0))) + ((eps * (-0.001388888888888889d0)) + (((eps * 0.5d0) - (eps * (-0.5d0))) * 0.041666666666666664d0))) - (eps * 0.08472222222222223d0)))) + (t_0 - (eps * 0.041666666666666664d0))))) - (eps * (-0.5d0))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = -0.5 * ((eps * -0.5) - (eps * 0.5));
	return eps + (Math.pow(x, 2.0) * (((eps * 0.5) + (Math.pow(x, 2.0) * (((eps * 0.20833333333333334) - (Math.pow(x, 2.0) * (((-0.5 * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - t_0))) + ((eps * -0.001388888888888889) + (((eps * 0.5) - (eps * -0.5)) * 0.041666666666666664))) - (eps * 0.08472222222222223)))) + (t_0 - (eps * 0.041666666666666664))))) - (eps * -0.5)));
}

Python

def code(x, eps):
	t_0 = -0.5 * ((eps * -0.5) - (eps * 0.5))
	return eps + (math.pow(x, 2.0) * (((eps * 0.5) + (math.pow(x, 2.0) * (((eps * 0.20833333333333334) - (math.pow(x, 2.0) * (((-0.5 * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - t_0))) + ((eps * -0.001388888888888889) + (((eps * 0.5) - (eps * -0.5)) * 0.041666666666666664))) - (eps * 0.08472222222222223)))) + (t_0 - (eps * 0.041666666666666664))))) - (eps * -0.5)))

Julia

function code(x, eps)
	t_0 = Float64(-0.5 * Float64(Float64(eps * -0.5) - Float64(eps * 0.5)))
	return Float64(eps + Float64((x ^ 2.0) * Float64(Float64(Float64(eps * 0.5) + Float64((x ^ 2.0) * Float64(Float64(Float64(eps * 0.20833333333333334) - Float64((x ^ 2.0) * Float64(Float64(Float64(-0.5 * Float64(Float64(eps * 0.20833333333333334) - Float64(Float64(eps * 0.041666666666666664) - t_0))) + Float64(Float64(eps * -0.001388888888888889) + Float64(Float64(Float64(eps * 0.5) - Float64(eps * -0.5)) * 0.041666666666666664))) - Float64(eps * 0.08472222222222223)))) + Float64(t_0 - Float64(eps * 0.041666666666666664))))) - Float64(eps * -0.5))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = -0.5 * ((eps * -0.5) - (eps * 0.5));
	tmp = eps + ((x ^ 2.0) * (((eps * 0.5) + ((x ^ 2.0) * (((eps * 0.20833333333333334) - ((x ^ 2.0) * (((-0.5 * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - t_0))) + ((eps * -0.001388888888888889) + (((eps * 0.5) - (eps * -0.5)) * 0.041666666666666664))) - (eps * 0.08472222222222223)))) + (t_0 - (eps * 0.041666666666666664))))) - (eps * -0.5)));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-0.5 * N[(N[(eps * -0.5), $MachinePrecision] - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(eps * 0.5), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(eps * 0.20833333333333334), $MachinePrecision] - N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(eps * 0.20833333333333334), $MachinePrecision] - N[(N[(eps * 0.041666666666666664), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * -0.001388888888888889), $MachinePrecision] + N[(N[(N[(eps * 0.5), $MachinePrecision] - N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.08472222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * -0.5), $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\left(0.5 \cdot \varepsilon + {x}^{2} \cdot \left(\left(0.20833333333333334 \cdot \varepsilon + {x}^{2} \cdot \left(0.08472222222222223 \cdot \varepsilon - \left(-0.5 \cdot \left(0.20833333333333334 \cdot \varepsilon - \left(-0.5 \cdot \left(0.5 \cdot \varepsilon - -0.5 \cdot \varepsilon\right) + 0.041666666666666664 \cdot \varepsilon\right)\right) + \left(-0.001388888888888889 \cdot \varepsilon + 0.041666666666666664 \cdot \left(0.5 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)\right) - \left(-0.5 \cdot \left(0.5 \cdot \varepsilon - -0.5 \cdot \varepsilon\right) + 0.041666666666666664 \cdot \varepsilon\right)\right)\right) - -0.5 \cdot \varepsilon\right)} \]

7. Final simplification 99.5%

   \[\leadsto \varepsilon + {x}^{2} \cdot \left(\left(\varepsilon \cdot 0.5 + {x}^{2} \cdot \left(\left(\varepsilon \cdot 0.20833333333333334 - {x}^{2} \cdot \left(\left(-0.5 \cdot \left(\varepsilon \cdot 0.20833333333333334 - \left(\varepsilon \cdot 0.041666666666666664 - -0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.5\right)\right)\right) + \left(\varepsilon \cdot -0.001388888888888889 + \left(\varepsilon \cdot 0.5 - \varepsilon \cdot -0.5\right) \cdot 0.041666666666666664\right)\right) - \varepsilon \cdot 0.08472222222222223\right)\right) + \left(-0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.5\right) - \varepsilon \cdot 0.041666666666666664\right)\right)\right) - \varepsilon \cdot -0.5\right) \]
8. Add Preprocessing

Alternative 8: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon - {x}^{2} \cdot \left(\varepsilon \cdot -0.5 - \left(\varepsilon \cdot 0.5 + {x}^{2} \cdot \left(\varepsilon \cdot 0.20833333333333334 - \left(\varepsilon \cdot 0.041666666666666664 - -0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.5\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  eps
  (*
   (pow x 2.0)
   (-
    (* eps -0.5)
    (+
     (* eps 0.5)
     (*
      (pow x 2.0)
      (-
       (* eps 0.20833333333333334)
       (-
        (* eps 0.041666666666666664)
        (* -0.5 (- (* eps -0.5) (* eps 0.5)))))))))))

C

double code(double x, double eps) {
	return eps - (pow(x, 2.0) * ((eps * -0.5) - ((eps * 0.5) + (pow(x, 2.0) * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - (-0.5 * ((eps * -0.5) - (eps * 0.5)))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps - ((x ** 2.0d0) * ((eps * (-0.5d0)) - ((eps * 0.5d0) + ((x ** 2.0d0) * ((eps * 0.20833333333333334d0) - ((eps * 0.041666666666666664d0) - ((-0.5d0) * ((eps * (-0.5d0)) - (eps * 0.5d0)))))))))
end function

Java

public static double code(double x, double eps) {
	return eps - (Math.pow(x, 2.0) * ((eps * -0.5) - ((eps * 0.5) + (Math.pow(x, 2.0) * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - (-0.5 * ((eps * -0.5) - (eps * 0.5)))))))));
}

Python

def code(x, eps):
	return eps - (math.pow(x, 2.0) * ((eps * -0.5) - ((eps * 0.5) + (math.pow(x, 2.0) * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - (-0.5 * ((eps * -0.5) - (eps * 0.5)))))))))

Julia

function code(x, eps)
	return Float64(eps - Float64((x ^ 2.0) * Float64(Float64(eps * -0.5) - Float64(Float64(eps * 0.5) + Float64((x ^ 2.0) * Float64(Float64(eps * 0.20833333333333334) - Float64(Float64(eps * 0.041666666666666664) - Float64(-0.5 * Float64(Float64(eps * -0.5) - Float64(eps * 0.5))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps - ((x ^ 2.0) * ((eps * -0.5) - ((eps * 0.5) + ((x ^ 2.0) * ((eps * 0.20833333333333334) - ((eps * 0.041666666666666664) - (-0.5 * ((eps * -0.5) - (eps * 0.5)))))))));
end

Wolfram

code[x_, eps_] := N[(eps - N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(eps * -0.5), $MachinePrecision] - N[(N[(eps * 0.5), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(eps * 0.20833333333333334), $MachinePrecision] - N[(N[(eps * 0.041666666666666664), $MachinePrecision] - N[(-0.5 * N[(N[(eps * -0.5), $MachinePrecision] - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\left(0.5 \cdot \varepsilon + {x}^{2} \cdot \left(0.20833333333333334 \cdot \varepsilon - \left(-0.5 \cdot \left(0.5 \cdot \varepsilon - -0.5 \cdot \varepsilon\right) + 0.041666666666666664 \cdot \varepsilon\right)\right)\right) - -0.5 \cdot \varepsilon\right)} \]

7. Final simplification 99.4%

   \[\leadsto \varepsilon - {x}^{2} \cdot \left(\varepsilon \cdot -0.5 - \left(\varepsilon \cdot 0.5 + {x}^{2} \cdot \left(\varepsilon \cdot 0.20833333333333334 - \left(\varepsilon \cdot 0.041666666666666664 - -0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.5\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 9: 98.2% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* (pow x 2.0) (- (* eps 0.5) (* eps -0.5)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] - N[(eps * -0.5), $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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Taylor expanded in x around 0 99.3%

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

7. Final simplification 99.3%

   \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot 0.5 - \varepsilon \cdot -0.5\right) \]
8. 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. 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. tan-quot 61.0%

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

   3. frac-sub 61.0%

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

4. Applied egg-rr 61.0%

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

5. Taylor expanded in eps around 0 99.6%

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

6. Taylor expanded in x around 0 98.7%

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

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 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)))