2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.6%

Time: 18.8s

Alternatives: 15

Speedup: 205.0×

• Could not converge on a ground truth

  1. x = -1.9869852053737772e-230
  2. eps = 7.900115087080797e-231

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: 62.6% 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

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

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
       (+
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (*
         eps
         (-
          0.3333333333333333
          (-
           (-
            (* -0.3333333333333333 (/ t_0 (/ (+ (cos (* x 2.0)) 1.0) 2.0)))
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
           t_1)))))
      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 * (((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / ((cos((x * 2.0)) + 1.0) / 2.0))) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))) - t_1))))) + 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 * (((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))) + (eps * (0.3333333333333333d0 - ((((-0.3333333333333333d0) * (t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0))) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0))) - t_1))))) + 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 * (((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / ((Math.cos((x * 2.0)) + 1.0) / 2.0))) - (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0))) - t_1))))) + 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 * (((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / ((math.cos((x * 2.0)) + 1.0) / 2.0))) - (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0))) - t_1))))) + 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(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))) + Float64(eps * Float64(0.3333333333333333 - Float64(Float64(Float64(-0.3333333333333333 * Float64(t_0 / Float64(Float64(cos(Float64(x * 2.0)) + 1.0) / 2.0))) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1))))) + 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 * (((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / ((cos((x * 2.0)) + 1.0) / 2.0))) - ((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1))))) + 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[(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 * N[(0.3333333333333333 - N[(N[(N[(-0.3333333333333333 * N[(t$95$0 / N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. add-cbrt-cube 25.0%

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

   2. pow1/3 24.8%

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

   3. pow3 24.8%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 24.8%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. tan-sum 25.1%

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

   2. div-inv 25.0%

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

   3. fmm-def 25.0%

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

6. Applied egg-rr 25.0%

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

7. Taylor expanded in eps around 0 99.9%

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

   1. unpow2 99.9%

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

   2. cos-mult 99.9%

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

9. Applied egg-rr 99.9%

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

    1. +-commutative 99.9%

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

    2. +-inverses 99.9%

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

    3. cos-0 99.9%

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

    4. count-2 99.9%

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

    5. *-commutative 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

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
       (+
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (*
         eps
         (-
          0.3333333333333333
          (-
           (-
            (* -0.3333333333333333 (/ t_0 (- 1.0 (pow x 2.0))))
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
           t_1)))))
      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 * (((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / (1.0 - pow(x, 2.0)))) - (pow(sin(x), 4.0) / pow(cos(x), 4.0))) - t_1))))) + 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 * (((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))) + (eps * (0.3333333333333333d0 - ((((-0.3333333333333333d0) * (t_0 / (1.0d0 - (x ** 2.0d0)))) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0))) - t_1))))) + 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 * (((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / (1.0 - Math.pow(x, 2.0)))) - (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0))) - t_1))))) + 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 * (((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / (1.0 - math.pow(x, 2.0)))) - (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0))) - t_1))))) + 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(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))) + Float64(eps * Float64(0.3333333333333333 - Float64(Float64(Float64(-0.3333333333333333 * Float64(t_0 / Float64(1.0 - (x ^ 2.0)))) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1))))) + 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 * (((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))) + (eps * (0.3333333333333333 - (((-0.3333333333333333 * (t_0 / (1.0 - (x ^ 2.0)))) - ((sin(x) ^ 4.0) / (cos(x) ^ 4.0))) - t_1))))) + 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[(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 * N[(0.3333333333333333 - N[(N[(N[(-0.3333333333333333 * N[(t$95$0 / N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. add-cbrt-cube 25.0%

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

   2. pow1/3 24.8%

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

   3. pow3 24.8%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 24.8%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. tan-sum 25.1%

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

   2. div-inv 25.0%

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

   3. fmm-def 25.0%

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

6. Applied egg-rr 25.0%

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

7. Taylor expanded in eps around 0 99.9%

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

8. Taylor expanded in x around 0 99.9%

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

   1. mul-1-neg 99.9%

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

   2. unsub-neg 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * ((t_0 / pow(cos(x), 2.0)) + ((eps * ((eps * 0.3333333333333333) + ((sin(x) * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + 1.0)) / cos(x)))) + 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
    code = eps * ((t_0 / (cos(x) ** 2.0d0)) + ((eps * ((eps * 0.3333333333333333d0) + ((sin(x) * ((t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)) + 1.0d0)) / cos(x)))) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. unpow2 99.9%

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

   2. cos-mult 99.9%

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

5. Applied egg-rr 99.9%

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

   1. +-commutative 99.9%

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

   2. +-inverses 99.9%

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

   3. cos-0 99.9%

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

   4. count-2 99.9%

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

   5. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * (((t_0 / pow(cos(x), 2.0)) + (eps * ((sin(x) * ((t_0 / ((cos((x * 2.0)) + 1.0) / 2.0)) + 1.0)) / cos(x)))) + 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
    code = eps * (((t_0 / (cos(x) ** 2.0d0)) + (eps * ((sin(x) * ((t_0 / ((cos((x * 2.0d0)) + 1.0d0) / 2.0d0)) + 1.0d0)) / cos(x)))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

   1. unpow2 99.9%

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

   2. cos-mult 99.9%

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

7. Applied egg-rr 99.6%

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

   1. +-commutative 99.9%

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

   2. +-inverses 99.9%

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

   3. cos-0 99.9%

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

   4. count-2 99.9%

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

   5. *-commutative 99.9%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

   1. add-cbrt-cube 25.0%

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

   2. pow1/3 24.8%

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

   3. pow3 24.8%

      \[\leadsto {\color{blue}{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 24.8%

   \[\leadsto \color{blue}{{\left({\left(\tan \left(x + \varepsilon\right) - \tan x\right)}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. tan-sum 25.1%

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

   2. div-inv 25.0%

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

   3. fmm-def 25.0%

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

6. Applied egg-rr 25.0%

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

7. Taylor expanded in eps around 0 99.9%

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

8. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 6: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. distribute-rgt-in 99.1%

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

   2. *-un-lft-identity 99.1%

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

   3. div-inv 99.1%

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

   4. pow-flip 99.1%

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

   5. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \varepsilon + \varepsilon \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) \]
9. 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 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.9%

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

   2. cos-mult 99.9%

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

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 99.9%

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

   2. +-inverses 99.9%

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

   3. cos-0 99.9%

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

   4. count-2 99.9%

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

   5. *-commutative 99.9%

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

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: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((0.3333333333333333 * pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * eps)) + 1.0))))) + 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 * ((1.3333333333333333d0 * (eps * eps)) + 1.0d0))))) + 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 * ((1.3333333333333333 * (eps * eps)) + 1.0))))) + 1.0);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((0.3333333333333333 * (eps ^ 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * eps)) + 1.0))))) + 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 + N[(x * N[(N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 98.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 9: 98.4% 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 61.0%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 98.4%

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

5. Taylor expanded in eps around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 10: 98.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right) + 1\right)\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (*
    x
    (+
     eps
     (*
      x
      (+ (* x (+ (* x 0.6666666666666666) (* eps 1.3333333333333333))) 1.0))))
   1.0)))

C

double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666d0) + (eps * 1.3333333333333333d0))) + 1.0d0)))) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))) + 1.0);
}

Python

def code(x, eps):
	return eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(x * Float64(eps + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.6666666666666666) + Float64(eps * 1.3333333333333333))) + 1.0)))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * (eps + (x * ((x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))) + 1.0)))) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * N[(eps + N[(x * N[(N[(x * N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(eps * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(0.6666666666666666 \cdot x + 0.8333333333333334 \cdot \varepsilon\right) - -0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate--l+ 98.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \color{blue}{\left(0.6666666666666666 \cdot x + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)}\right)\right)\right) \]

   2. *-commutative 98.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\color{blue}{x \cdot 0.6666666666666666} + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)\right) \]

   3. distribute-rgt-out-- 98.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \color{blue}{\varepsilon \cdot \left(0.8333333333333334 - -0.5\right)}\right)\right)\right)\right) \]

   4. metadata-eval 98.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot \color{blue}{1.3333333333333333}\right)\right)\right)\right) \]

8. Simplified 98.3%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right)\right)\right)}\right) \]

9. Final simplification 98.3%

   \[\leadsto \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right) + 1\right)\right) + 1\right) \]
10. Add Preprocessing

Alternative 11: 98.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * ((x * (eps + (x * ((x * (eps * 1.3333333333333333)) + 1.0)))) + 1.0);
}

Python

def code(x, eps):
	return eps * ((x * (eps + (x * ((x * (eps * 1.3333333333333333)) + 1.0)))) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 98.2%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. distribute-rgt-out-- 98.2%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \color{blue}{\left(\varepsilon \cdot \left(0.8333333333333334 - -0.5\right)\right)}\right)\right)\right) \]

   2. metadata-eval 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

   \[\leadsto \varepsilon \cdot \left(x \cdot \left(\varepsilon + x \cdot \left(x \cdot \left(\varepsilon \cdot 1.3333333333333333\right) + 1\right)\right) + 1\right) \]
10. Add Preprocessing

Alternative 12: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. unpow2 98.2%

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

   3. distribute-lft-out 98.2%

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

8. Simplified 98.2%

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

Alternative 13: 97.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 97.9%

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

   1. distribute-rgt-in 97.9%

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

   2. *-un-lft-identity 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

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

Alternative 14: 97.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * ((eps * x) + 1.0);
}

Python

def code(x, eps):
	return eps * ((eps * x) + 1.0)

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * x), $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 99.5%

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

   1. associate--l+ 99.6%

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

   2. associate-/l* 99.6%

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

   3. mul-1-neg 99.6%

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

   4. mul-1-neg 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 97.9%

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

7. Final simplification 97.9%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot x + 1\right) \]
8. Add Preprocessing

Alternative 15: 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 61.0%

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

3. Taylor expanded in x around 0 97.9%

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

4. Taylor expanded in eps around 0 97.9%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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]

Developer Target 2: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024185 
(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)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

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